Intermediate Algebra 2e

# Review Exercises

Intermediate Algebra 2eReview Exercises

### Review Exercises

##### Solve Quadratic Equations Using the Square Root Property

Solve Quadratic Equations of the form ax2 = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

395.

$y 2 = 144 y 2 = 144$

396.

$n 2 − 80 = 0 n 2 − 80 = 0$

397.

$4 a 2 = 100 4 a 2 = 100$

398.

$2 b 2 = 72 2 b 2 = 72$

399.

$r 2 + 32 = 0 r 2 + 32 = 0$

400.

$t 2 + 18 = 0 t 2 + 18 = 0$

401.

$2 3 w 2 − 20 = 30 2 3 w 2 − 20 = 30$

402.

11. $5c2+3=195c2+3=19$

Solve Quadratic Equations of the Form $a(x−h)2=ka(x−h)2=k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

403.

$( p − 5 ) 2 + 3 = 19 ( p − 5 ) 2 + 3 = 19$

404.

$( u + 1 ) 2 = 45 ( u + 1 ) 2 = 45$

405.

$( x − 1 4 ) 2 = 3 16 ( x − 1 4 ) 2 = 3 16$

406.

$( y − 2 3 ) 2 = 2 9 ( y − 2 3 ) 2 = 2 9$

407.

$( n − 4 ) 2 − 50 = 150 ( n − 4 ) 2 − 50 = 150$

408.

$( 4 c − 1 ) 2 = −18 ( 4 c − 1 ) 2 = −18$

409.

$n 2 + 10 n + 25 = 12 n 2 + 10 n + 25 = 12$

410.

$64 a 2 + 48 a + 9 = 81 64 a 2 + 48 a + 9 = 81$

##### Solve Quadratic Equations by Completing the Square

Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

411.

$x 2 + 22 x x 2 + 22 x$

412.

$m 2 − 8 m m 2 − 8 m$

413.

$a 2 − 3 a a 2 − 3 a$

414.

$b 2 + 13 b b 2 + 13 b$

In the following exercises, solve by completing the square.

415.

$d 2 + 14 d = −13 d 2 + 14 d = −13$

416.

$y 2 − 6 y = 36 y 2 − 6 y = 36$

417.

$m 2 + 6 m = −109 m 2 + 6 m = −109$

418.

$t 2 − 12 t = −40 t 2 − 12 t = −40$

419.

$v 2 − 14 v = −31 v 2 − 14 v = −31$

420.

$w 2 − 20 w = 100 w 2 − 20 w = 100$

421.

$m 2 + 10 m − 4 = −13 m 2 + 10 m − 4 = −13$

422.

$n 2 − 6 n + 11 = 34 n 2 − 6 n + 11 = 34$

423.

$a 2 = 3 a + 8 a 2 = 3 a + 8$

424.

$b 2 = 11 b − 5 b 2 = 11 b − 5$

425.

$( u + 8 ) ( u + 4 ) = 14 ( u + 8 ) ( u + 4 ) = 14$

426.

$( z − 10 ) ( z + 2 ) = 28 ( z − 10 ) ( z + 2 ) = 28$

Solve Quadratic Equations of the form ax2 + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

427.

$3 p 2 − 18 p + 15 = 15 3 p 2 − 18 p + 15 = 15$

428.

$5 q 2 + 70 q + 20 = 0 5 q 2 + 70 q + 20 = 0$

429.

$4 y 2 − 6 y = 4 4 y 2 − 6 y = 4$

430.

$2 x 2 + 2 x = 4 2 x 2 + 2 x = 4$

431.

$3 c 2 + 2 c = 9 3 c 2 + 2 c = 9$

432.

$4 d 2 − 2 d = 8 4 d 2 − 2 d = 8$

433.

$2 x 2 + 6 x = −5 2 x 2 + 6 x = −5$

434.

$2 x 2 + 4 x = −5 2 x 2 + 4 x = −5$

In the following exercises, solve by using the Quadratic Formula.

435.

$4 x 2 − 5 x + 1 = 0 4 x 2 − 5 x + 1 = 0$

436.

$7 y 2 + 4 y − 3 = 0 7 y 2 + 4 y − 3 = 0$

437.

$r 2 − r − 42 = 0 r 2 − r − 42 = 0$

438.

$t 2 + 13 t + 22 = 0 t 2 + 13 t + 22 = 0$

439.

$4 v 2 + v − 5 = 0 4 v 2 + v − 5 = 0$

440.

$2 w 2 + 9 w + 2 = 0 2 w 2 + 9 w + 2 = 0$

441.

$3 m 2 + 8 m + 2 = 0 3 m 2 + 8 m + 2 = 0$

442.

$5 n 2 + 2 n − 1 = 0 5 n 2 + 2 n − 1 = 0$

443.

$6 a 2 − 5 a + 2 = 0 6 a 2 − 5 a + 2 = 0$

444.

$4 b 2 − b + 8 = 0 4 b 2 − b + 8 = 0$

445.

$u ( u − 10 ) + 3 = 0 u ( u − 10 ) + 3 = 0$

446.

$5 z ( z − 2 ) = 3 5 z ( z − 2 ) = 3$

447.

$1 8 p 2 − 1 5 p = − 1 20 1 8 p 2 − 1 5 p = − 1 20$

448.

$2 5 q 2 + 3 10 q = 1 10 2 5 q 2 + 3 10 q = 1 10$

449.

$4 c 2 + 4 c + 1 = 0 4 c 2 + 4 c + 1 = 0$

450.

$9 d 2 − 12 d = −4 9 d 2 − 12 d = −4$

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions for each quadratic equation.

451.

$9x2−6x+1=09x2−6x+1=0$
$3y2−8y+1=03y2−8y+1=0$
$7m2+12m+4=07m2+12m+4=0$
$5n2−n+1=05n2−n+1=0$

452.

$5x2−7x−8=05x2−7x−8=0$
$7x2−10x+5=07x2−10x+5=0$
$25x2−90x+81=025x2−90x+81=0$
$15x2−8x+4=015x2−8x+4=0$

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

453.

$16r2−8r+1=016r2−8r+1=0$
$5t2−8t+3=95t2−8t+3=9$
$3(c+2)2=153(c+2)2=15$

454.

$4d2+10d−5=214d2+10d−5=21$
$25x2−60x+36=025x2−60x+36=0$
$6(5v−7)2=1506(5v−7)2=150$

##### Solve Equations in Quadratic Form

In the following exercises, solve.

455.

$x 4 − 14 x 2 + 24 = 0 x 4 − 14 x 2 + 24 = 0$

456.

$x 4 + 4 x 2 − 32 = 0 x 4 + 4 x 2 − 32 = 0$

457.

$4 x 4 − 5 x 2 + 1 = 0 4 x 4 − 5 x 2 + 1 = 0$

458.

$( 2 y + 3 ) 2 + 3 ( 2 y + 3 ) − 28 = 0 ( 2 y + 3 ) 2 + 3 ( 2 y + 3 ) − 28 = 0$

459.

$x + 3 x − 28 = 0 x + 3 x − 28 = 0$

460.

$6 x + 5 x − 6 = 0 6 x + 5 x − 6 = 0$

461.

$x 2 3 − 10 x 1 3 + 24 = 0 x 2 3 − 10 x 1 3 + 24 = 0$

462.

$x + 7 x 1 2 + 6 = 0 x + 7 x 1 2 + 6 = 0$

463.

$8 x −2 − 2 x −1 − 3 = 0 8 x −2 − 2 x −1 − 3 = 0$

##### Solve Applications of Quadratic Equations

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

464.

Find two consecutive odd numbers whose product is 323.

465.

Find two consecutive even numbers whose product is 624.

466.

A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.

467.

Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.

468.

A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.

469.

A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

470.

The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

471.

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.

472.

A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula h = −16t2 + v0t to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

473.

The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the wind that affected the plane?

474.

Ezra kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current?

475.

Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

##### Graph Quadratic Functions Using Properties

Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

476.

Graph $y=x2−2y=x2−2$

477.

Graph $y=−x2+3y=−x2+3$

In the following exercises, determine if the following parabolas open up or down.

478.

$y=−3x2+3x−1y=−3x2+3x−1$
$y=5x2+6x+3y=5x2+6x+3$

479.

$y=x2+8x−1y=x2+8x−1$
$y=−4x2−7x+1y=−4x2−7x+1$

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find the equation of the axis of symmetry and the vertex.

480.

$y = − x 2 + 6 x + 8 y = − x 2 + 6 x + 8$

481.

$y = 2 x 2 − 8 x + 1 y = 2 x 2 − 8 x + 1$

Find the Intercepts of a Parabola

In the following exercises, find the x- and y-intercepts.

482.

$y = x 2 − 4 x − 5 y = x 2 − 4 x − 5$

483.

$y = x 2 − 8 x + 15 y = x 2 − 8 x + 15$

484.

$y = x 2 − 4 x + 10 y = x 2 − 4 x + 10$

485.

$y = −5 x 2 − 30 x − 46 y = −5 x 2 − 30 x − 46$

486.

$y = 16 x 2 − 8 x + 1 y = 16 x 2 − 8 x + 1$

487.

$y = x 2 + 16 x + 64 y = x 2 + 16 x + 64$

In the following exercises, graph by using its properties.

488.

$y = x 2 + 8 x + 15 y = x 2 + 8 x + 15$

489.

$y = x 2 − 2 x − 3 y = x 2 − 2 x − 3$

490.

$y = − x 2 + 8 x − 16 y = − x 2 + 8 x − 16$

491.

$y = 4 x 2 − 4 x + 1 y = 4 x 2 − 4 x + 1$

492.

$y = x 2 + 6 x + 13 y = x 2 + 6 x + 13$

493.

$y = −2 x 2 − 8 x − 12 y = −2 x 2 − 8 x − 12$

Solve Maximum and Minimum Applications

In the following exercises, find the minimum or maximum value.

494.

$y = 7 x 2 + 14 x + 6 y = 7 x 2 + 14 x + 6$

495.

$y = −3 x 2 + 12 x − 10 y = −3 x 2 + 12 x − 10$

In the following exercises, solve. Rounding answers to the nearest tenth.

496.

A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation h = −16t2 + 112t to find how long it will take the ball to reach maximum height, and then find the maximum height.

497.

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation $A(x)=x90−x2A(x)=x90−x2$ gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

##### Graph Quadratic Functions Using Transformations

Graph Quadratic Functions of the form $f(x)=x2+kf(x)=x2+k$

In the following exercises, graph each function using a vertical shift.

498.

$g ( x ) = x 2 + 4 g ( x ) = x 2 + 4$

499.

$h ( x ) = x 2 − 3 h ( x ) = x 2 − 3$

In the following exercises, graph each function using a horizontal shift.

500.

$f ( x ) = ( x + 1 ) 2 f ( x ) = ( x + 1 ) 2$

501.

$g ( x ) = ( x − 3 ) 2 g ( x ) = ( x − 3 ) 2$

In the following exercises, graph each function using transformations.

502.

$f ( x ) = ( x + 2 ) 2 + 3 f ( x ) = ( x + 2 ) 2 + 3$

503.

$f ( x ) = ( x + 3 ) 2 − 2 f ( x ) = ( x + 3 ) 2 − 2$

504.

$f ( x ) = ( x − 1 ) 2 + 4 f ( x ) = ( x − 1 ) 2 + 4$

505.

$f ( x ) = ( x − 4 ) 2 − 3 f ( x ) = ( x − 4 ) 2 − 3$

Graph Quadratic Functions of the form $f(x)=ax2f(x)=ax2$

In the following exercises, graph each function.

506.

$f ( x ) = 2 x 2 f ( x ) = 2 x 2$

507.

$f ( x ) = − x 2 f ( x ) = − x 2$

508.

$f ( x ) = 1 2 x 2 f ( x ) = 1 2 x 2$

In the following exercises, rewrite each function in the $f(x)=a(x−h)2+kf(x)=a(x−h)2+k$ form by completing the square.

509.

$f ( x ) = 2 x 2 − 4 x − 4 f ( x ) = 2 x 2 − 4 x − 4$

510.

$f ( x ) = 3 x 2 + 12 x + 8 f ( x ) = 3 x 2 + 12 x + 8$

In the following exercises, rewrite each function in $f(x)=a(x−h)2+kf(x)=a(x−h)2+k$ form and graph it by using transformations.

511.

$f ( x ) = 3 x 2 − 6 x − 1 f ( x ) = 3 x 2 − 6 x − 1$

512.

$f ( x ) = −2 x 2 − 12 x − 5 f ( x ) = −2 x 2 − 12 x − 5$

513.

$f ( x ) = 2 x 2 + 4 x + 6 f ( x ) = 2 x 2 + 4 x + 6$

514.

$f ( x ) = 3 x 2 − 12 x + 7 f ( x ) = 3 x 2 − 12 x + 7$

In the following exercises, rewrite each function in $f(x)=a(x−h)2+kf(x)=a(x−h)2+k$ form and graph it using properties.

515.

$f ( x ) = −3 x 2 − 12 x − 5 f ( x ) = −3 x 2 − 12 x − 5$

516.

$f ( x ) = 2 x 2 − 12 x + 7 f ( x ) = 2 x 2 − 12 x + 7$

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f(x)=a(x−h)2+kf(x)=a(x−h)2+k$ form.

517.
518.

In the following exercises, solve graphically and write the solution in interval notation.

519.

$x 2 − x − 6 > 0 x 2 − x − 6 > 0$

520.

$x 2 + 4 x + 3 ≤ 0 x 2 + 4 x + 3 ≤ 0$

521.

$− x 2 − x + 2 ≥ 0 − x 2 − x + 2 ≥ 0$

522.

$− x 2 + 2 x + 3 < 0 − x 2 + 2 x + 3 < 0$

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

523.

$x 2 − 6 x + 8 < 0 x 2 − 6 x + 8 < 0$

524.

$x 2 + x > 12 x 2 + x > 12$

525.

$x 2 − 6 x + 4 ≤ 0 x 2 − 6 x + 4 ≤ 0$

526.

$2 x 2 + 7 x − 4 > 0 2 x 2 + 7 x − 4 > 0$

527.

$− x 2 + x − 6 > 0 − x 2 + x − 6 > 0$

528.

$x 2 − 2 x + 4 ≥ 0 x 2 − 2 x + 4 ≥ 0$

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