### Key Concepts

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

- Square Root Property
- If ${x}^{2}=k$, then $x=\sqrt{k}\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}x=\text{\u2212}\sqrt{k}$ or $x=\pm \sqrt{k}$

How to solve a quadratic equation using the square root property.- Step 1. Isolate the quadratic term and make its coefficient one.
- Step 2. Use Square Root Property.
- Step 3. Simplify the radical.
- Step 4. Check the solutions.

### 9.2 Solve Quadratic Equations by Completing the Square

- Binomial Squares Pattern

If*a*and*b*are real numbers,

- How to Complete a Square
- Step 1. Identify
*b*, the coefficient of*x*. - Step 2. Find ${\left(\frac{1}{2}b\right)}^{2},$ the number to complete the square.
- Step 3. Add the ${\left(\frac{1}{2}b\right)}^{2}$ to
*x*^{2}+*bx* - Step 4. Rewrite the trinomial as a binomial square

- Step 1. Identify
- How to solve a quadratic equation of the form
*ax*^{2}+*bx*+*c*= 0 by completing the square.- Step 1. Divide by
*a*to make the coefficient of*x*^{2}term 1. - Step 2. Isolate the variable terms on one side and the constant terms on the other.
- Step 3. Find ${\left(\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}b\right)}^{2},$ the number needed to complete the square. Add it to both sides of the equation.
- Step 4. Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right.
- Step 5. Use the Square Root Property.
- Step 6. Simplify the radical and then solve the two resulting equations.
- Step 7. Check the solutions.

- Step 1. Divide by

### 9.3 Solve Quadratic Equations Using the Quadratic Formula

- Quadratic Formula
- The solutions to a quadratic equation of the form
*ax*^{2}+*bx*+*c*= 0, $a\ne 0$ are given by the formula:

$$x=\frac{\text{\u2212}b\pm \sqrt{{b}^{2}-4ac}}{2a}$$

- The solutions to a quadratic equation of the form
- How to solve a quadratic equation using the Quadratic Formula.
- Step 1. Write the quadratic equation in standard form,
*ax*^{2}+*bx*+*c*= 0. Identify the values of*a*,*b*,*c*. - Step 2. Write the Quadratic Formula. Then substitute in the values of
*a*,*b*,*c*. - Step 3. Simplify.
- Step 4. Check the solutions.

- Step 1. Write the quadratic equation in standard form,
- Using the Discriminant,
*b*^{2}− 4*ac*, to Determine the Number and Type of Solutions of a Quadratic Equation- For a quadratic equation of the form
*ax*^{2}+*bx*+*c*= 0, $a\ne 0,$- If
*b*^{2}− 4*ac*> 0, the equation has 2 real solutions. - if
*b*^{2}− 4*ac*= 0, the equation has 1 real solution. - if
*b*^{2}− 4*ac*< 0, the equation has 2 complex solutions.

- If

- For a quadratic equation of the form
- Methods to Solve Quadratic Equations:
- Factoring
- Square Root Property
- Completing the Square
- Quadratic Formula

- How to identify the most appropriate method to solve a quadratic equation.
- Step 1. Try Factoring first. If the quadratic factors easily, this method is very quick.
- Step 2. Try the
**Square Root Property**next. If the equation fits the form*ax*^{2}=*k*or*a*(*x*−*h*)^{2}=*k*, it can easily be solved by using the Square Root Property. - Step 3. Use the
**Quadratic Formula.**Any other quadratic equation is best solved by using the Quadratic Formula.

### 9.4 Solve Quadratic Equations in Quadratic Form

- How to solve equations in quadratic form.
- Step 1. Identify a substitution that will put the equation in quadratic form.
- Step 2. Rewrite the equation with the substitution to put it in quadratic form.
- Step 3. Solve the quadratic equation for
*u*. - Step 4. Substitute the original variable back into the results, using the substitution.
- Step 5. Solve for the original variable.
- Step 6. Check the solutions.

### 9.5 Solve Applications of Quadratic Equations

- Methods to Solve Quadratic Equations
- Factoring
- Square Root Property
- Completing the Square
- Quadratic Formula

- How to use a Problem-Solving Strategy.
- Step 1.
**Read**the problem. Make sure all the words and ideas are understood. - Step 2.
**Identify**what we are looking for. - Step 3.
**Name**what we are looking for. Choose a variable to represent that quantity. - Step 4.
**Translate**into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. - Step 5.
**Solve**the equation using good algebra techniques. - Step 6.
**Check**the answer in the problem and make sure it makes sense. - Step 7.
**Answer**the question with a complete sentence.

- Step 1.
- Area of a Triangle
- For a triangle with base,
*b*, and height,*h*, the area,*A*, is given by the formula $A=\frac{1}{2}bh.$

- For a triangle with base,
- Area of a Rectangle
- For a rectangle with length,
*L*, and width,*W*, the area,*A*, is given by the formula*A*=*LW*.

- For a rectangle with length,
- Pythagorean Theorem
- In any right triangle, where
*a*and*b*are the lengths of the legs, and*c*is the length of the hypotenuse,*a*^{2}+*b*^{2}=*c*^{2}.

- In any right triangle, where
- Projectile motion
- The height in feet,
*h*, of an object shot upwards into the air with initial velocity,*v*_{0}, after*t*seconds is given by the formula*h*= −16*t*^{2}+*v*_{0}*t*.

- The height in feet,

### 9.6 Graph Quadratic Functions Using Properties

- Parabola Orientation
- For the graph of the quadratic function $f\left(x\right)=a{x}^{2}+bx+c,$ if
*a*> 0, the parabola opens upward.*a*< 0, the parabola opens downward.

- For the graph of the quadratic function $f\left(x\right)=a{x}^{2}+bx+c,$ if
- Axis of Symmetry and Vertex of a Parabola The graph of the function $f\left(x\right)=a{x}^{2}+bx+c$ is a parabola where:
- the axis of symmetry is the vertical line $x=-\frac{b}{2a}.$
- the vertex is a point on the axis of symmetry, so its
*x*-coordinate is $-\frac{b}{2a}.$ - the
*y*-coordinate of the vertex is found by substituting $x=-\frac{b}{2a}$ into the quadratic equation.

- Find the Intercepts of a Parabola
- To find the intercepts of a parabola whose function is $f\left(x\right)=a{x}^{2}+bx+c:$

$$\begin{array}{cccccc}\hfill \mathit{\text{y}}\mathbf{\text{-intercept}}\hfill & & & & & \hfill \mathit{\text{x}}\mathbf{\text{-intercepts}}\hfill \\ \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}x=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}f\left(x\right).\hfill & & & & & \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}f\left(x\right)=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}x.\hfill \end{array}$$

- To find the intercepts of a parabola whose function is $f\left(x\right)=a{x}^{2}+bx+c:$
- How to graph a quadratic function using properties.
- Step 1. Determine whether the parabola opens upward or downward.
- Step 2. Find the equation of the axis of symmetry.
- Step 3. Find the vertex.
- Step 4. Find the
*y*-intercept. Find the point symmetric to the*y*-intercept across the axis of symmetry. - Step 5. Find the
*x*-intercepts. Find additional points if needed. - Step 6. Graph the parabola.

- Minimum or Maximum Values of a Quadratic Equation
- The
*y*-coordinate of the vertex of the graph of a quadratic equation is the *minimum*value of the quadratic equation if the parabola opens*upward*.*maximum*value of the quadratic equation if the parabola opens*downward*.

- The

### 9.7 Graph Quadratic Functions Using Transformations

- Graph a Quadratic Function of the form $f(x)={x}^{2}+k$ Using a Vertical Shift
- The graph of $f(x)={x}^{2}+k$ shifts the graph of $f\left(x\right)={x}^{2}$ vertically k units.
- If
*k*> 0, shift the parabola vertically up*k*units. - If
*k*< 0, shift the parabola vertically down $\left|k\right|$ units.

- If

- The graph of $f(x)={x}^{2}+k$ shifts the graph of $f\left(x\right)={x}^{2}$ vertically k units.
- Graph a Quadratic Function of the form $f(x)={\left(x-h\right)}^{2}$ Using a Horizontal Shift
- The graph of $f(x)={\left(x-h\right)}^{2}$ shifts the graph of $f\left(x\right)={x}^{2}$ horizontally h units.
- If
*h*> 0, shift the parabola horizontally left*h*units. - If
*h*< 0, shift the parabola horizontally right $\left|h\right|$ units.

- If

- The graph of $f(x)={\left(x-h\right)}^{2}$ shifts the graph of $f\left(x\right)={x}^{2}$ horizontally h units.
- Graph of a Quadratic Function of the form $f(x)=a{x}^{2}$
- The coefficient
*a*in the function $f(x)=a{x}^{2}$ affects the graph of $f\left(x\right)={x}^{2}$ by stretching or compressing it.

If $0<\left|a\right|<1,$ then the graph of $f(x)=a{x}^{2}$ will be “wider” than the graph of $f\left(x\right)={x}^{2}.$

If $\left|a\right|>1,$ then the graph of $f(x)=a{x}^{2}$ will be “skinnier” than the graph of $f\left(x\right)={x}^{2}.$

- The coefficient
- How to graph a quadratic function using transformations
- Step 1. Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.
- Step 2. Graph the function using transformations.

- Graph a quadratic function in the vertex form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ using properties
- Step 1. Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form.
- Step 2. Determine whether the parabola opens upward,
*a*> 0, or downward, a < 0. - Step 3. Find the axis of symmetry,
*x*=*h*. - Step 4. Find the vertex, (
*h*,*k*). - Step 5. Find the
*y*-intercept. Find the point symmetric to the*y*-intercept across the axis of symmetry. - Step 6. Find the
*x*-intercepts, if possible. - Step 7. Graph the parabola.

### 9.8 Solve Quadratic Inequalities

- Solve a Quadratic Inequality Graphically
- Step 1. Write the quadratic inequality in standard form.
- Step 2. Graph the function $f\left(x\right)=a{x}^{2}+bx+c$ using properties or transformations.
- Step 3. Determine the solution from the graph.

- How to Solve a Quadratic Inequality Algebraically
- Step 1. Write the quadratic inequality in standard form.
- Step 2. Determine the critical points -- the solutions to the related quadratic equation.
- Step 3. Use the critical points to divide the number line into intervals.
- Step 4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.
- Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.