Ch. 10 Review Exercises - Elementary Algebra 2e

Review Exercises

10.1 Solve Quadratic Equations Using the Square Root Property

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

213.

$x^{2} = 100$

214.

$y^{2} = 144$

215.

$m^{2} - 40 = 0$

216.

$n^{2} - 80 = 0$

217.

$4 a^{2} = 100$

218.

$2 b^{2} = 72$

219.

$r^{2} + 32 = 0$

220.

$t^{2} + 18 = 0$

221.

$\frac{4}{3} v^{2} + 4 = 28$

222.

$\frac{2}{3} w^{2} - 20 = 30$

223.

$5 c^{2} + 3 = 19$

224.

$3 d^{2} - 6 = 43$

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

225.

${(p - 5)}^{2} + 3 = 19$

226.

${(q + 4)}^{2} = 9$

227.

${(u + 1)}^{2} = 45$

228.

${(z - 5)}^{2} = 50$

229.

${(x - \frac{1}{4})}^{2} = \frac{3}{16}$

230.

${(y - \frac{2}{3})}^{2} = \frac{2}{9}$

231.

${(m - 7)}^{2} + 6 = 30$

232.

${(n - 4)}^{2} - 50 = 150$

233.

${(5 c + 3)}^{2} = −20$

234.

${(4 c - 1)}^{2} = −18$

235.

$m^{2} - 6 m + 9 = 48$

236.

$n^{2} + 10 n + 25 = 12$

237.

$64 a^{2} + 48 a + 9 = 81$

238.

$4 b^{2} - 28 b + 49 = 25$

10.2 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.

239.

$x^{2} + 22 x$

240.

$y^{2} + 6 y$

241.

$m^{2} - 8 m$

242.

$n^{2} - 10 n$

243.

$a^{2} - 3 a$

244.

$b^{2} + 13 b$

245.

$p^{2} + \frac{4}{5} p$

246.

$q^{2} - \frac{1}{3} q$

In the following exercises, solve by completing the square.

247.

$c^{2} + 20 c = 21$

248.

$d^{2} + 14 d = −13$

249.

$x^{2} - 4 x = 32$

250.

$y^{2} - 16 y = 36$

251.

$r^{2} + 6 r = −100$

252.

$t^{2} - 12 t = −40$

253.

$v^{2} - 14 v = −31$

254.

$w^{2} - 20 w = 100$

255.

$m^{2} + 10 m - 4 = −13$

256.

$n^{2} - 6 n + 11 = 34$

257.

$a^{2} = 3 a + 8$

258.

$b^{2} = 11 b - 5$

259.

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

260.

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

261.

$3 p^{2} - 18 p + 15 = 15$

262.

$5 q^{2} + 70 q + 20 = 0$

263.

$4 y^{2} - 6 y = 4$

264.

$2 x^{2} + 2 x = 4$

265.

$3 c^{2} + 2 c = 9$

266.

$4 d^{2} - 2 d = 8$

10.3 Solve Quadratic Equations Using the Quadratic Formula

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

267.

$4 x^{2} - 5 x + 1 = 0$

268.

$7 y^{2} + 4 y - 3 = 0$

269.

$r^{2} - r - 42 = 0$

270.

$t^{2} + 13 t + 22 = 0$

271.

$4 v^{2} + v - 5 = 0$

272.

$2 w^{2} + 9 w + 2 = 0$

273.

$3 m^{2} + 8 m + 2 = 0$

274.

$5 n^{2} + 2 n - 1 = 0$

275.

$6 a^{2} - 5 a + 2 = 0$

276.

$4 b^{2} - b + 8 = 0$

277.

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

278.

$5 z (z - 2) = 3$

279.

$\frac{1}{8} p^{2} - \frac{1}{5} p = - \frac{1}{20}$

280.

$\frac{2}{5} q^{2} + \frac{3}{10} q = \frac{1}{10}$

281.

$4 c^{2} + 4 c + 1 = 0$

282.

$9 d^{2} - 12 d = −4$

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

283.

ⓐ $9 x^{2} - 6 x + 1 = 0$
ⓑ $3 y^{2} - 8 y + 1 = 0$
ⓒ $7 m^{2} + 12 m + 4 = 0$
ⓓ $5 n^{2} - n + 1 = 0$

284.

ⓐ $5 x^{2} - 7 x - 8 = 0$
ⓑ $7 x^{2} - 10 x + 5 = 0$
ⓒ $25 x^{2} - 90 x + 81 = 0$
ⓓ $15 x^{2} - 8 x + 4 = 0$

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.

285.

ⓐ $16 r^{2} - 8 r + 1 = 0$
ⓑ $5 t^{2} - 8 t + 3 = 9$
ⓒ $3 {(c + 2)}^{2} = 15$

286.

ⓐ $4 d^{2} + 10 d - 5 = 21$
ⓑ $25 x^{2} - 60 x + 36 = 0$
ⓒ $6 {(5 v - 7)}^{2} = 150$

10.4 Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.

287.

Find two consecutive odd numbers whose product is 323.

288.

Find two consecutive even numbers whose product is 624.

289.

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.

290.

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.

291.

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.

The image shows a rectangular pathway with a right inlaid in the lower left corner. The right angle of the triangle overlays the lower left corner of the rectangle. The left leg of the right triangle overlays the left side of the rectangle and the hypotenuse of the right triangle runs from the upper left corner of the rectangle to a point on the bottom of the rectangle.

292.

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.

293.

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.

294.

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a 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.

The image shows four rectangular tables placed side by side to create one large table.

295.

A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula $h = −16 t^{2} + v_{0} t$ to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

296.

A bullet is fired straight up from the ground at a velocity of 320 ft/sec. Use the formula $h = −16 t^{2} + v_{0} t$ to determine when the bullet will reach 800 feet. Round to the nearest tenth.

10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.

297.

Graph $y = x^{2} - 2$

298.

Graph $y = − x^{2} + 3$

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

299.

$y = −3 x^{2} + 3 x - 1$

300.

$y = 5 x^{2} + 6 x + 3$

301.

$y = x^{2} + 8 x - 1$

302.

$y = −4 x^{2} - 7 x + 1$

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

303.

$y = − x^{2} + 6 x + 8$

304.

$y = 2 x^{2} - 8 x + 1$

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

305.

$y = x^{2} - 4 x - 5$

306.

$y = x^{2} - 8 x + 15$

307.

$y = x^{2} - 4 x + 10$

308.

$y = −5 x^{2} - 30 x - 46$

309.

$y = 16 x^{2} - 8 x + 1$

310.

$y = x^{2} + 16 x + 64$

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

311.

$y = x^{2} + 8 x + 15$

312.

$y = x^{2} - 2 x - 3$

313.

$y = − x^{2} + 8 x - 16$

314.

$y = 4 x^{2} - 4 x + 1$

315.

$y = x^{2} + 6 x + 13$

316.

$y = −2 x^{2} - 8 x - 12$

317.

$y = −4 x^{2} + 16 x - 11$

318.

$y = x^{2} + 8 x + 10$

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

319.

$y = 7 x^{2} + 14 x + 6$

320.

$y = −3 x^{2} + 12 x - 10$

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

321.

A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation $h = −16 t^{2} + 112 t$ to find how long it will take the ball to reach maximum height, and then find the maximum height.

322.

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 $A = −2 x^{2} + 180 x$ 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.