### Key Concepts

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

- Square Root Property

If ${x}^{2}=k$, and $k\xe2\u2030\yen 0$, then $x=\sqrt{k}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x=\text{\xe2\u02c6\u2019}\sqrt{k}$.

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

- Binomial Squares Pattern If $a,b$ are real numbers,

${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$

${\left(a\xe2\u02c6\u2019b\right)}^{2}={a}^{2}\xe2\u02c6\u20192ab+{b}^{2}$

- Complete a Square

To complete the square of ${x}^{2}+bx$:- 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 ${x}^{2}+bx$.

#### 10.3 Solve Quadratic Equations Using the Quadratic Formula

**Quadratic Formula**The solutions to a quadratic equation of the form $a{x}^{2}+bx+c=0,$ $a\xe2\u20300$ are given by the formula:

$$x=\frac{\text{\xe2\u02c6\u2019}b\xc2\pm \sqrt{{b}^{2}\xe2\u02c6\u20194ac}}{2a}$$**Solve a Quadratic Equation Using the Quadratic Formula**

To solve a quadratic equation using the Quadratic Formula.- Step 1. Write the quadratic formula in standard form. Identify the $a,b,c$ values.
- Step 2. Write the quadratic formula. Then substitute in the values of $a,b,c.$
- Step 3. Simplify.
- Step 4. Check the solutions.

**Using the Discriminant, ${b}^{2}\xe2\u02c6\u20194ac$, to Determine the Number of Solutions of a Quadratic Equation**

For a quadratic equation of the form $a{x}^{2}+bx+c=0,$ $a\xe2\u20300,$- if ${b}^{2}\xe2\u02c6\u20194ac>0$, the equation has 2 solutions.
- if ${b}^{2}\xe2\u02c6\u20194ac=0$, the equation has 1 solution.
- if ${b}^{2}\xe2\u02c6\u20194ac<0$, the equation has no real solutions.

**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 $a{x}^{2}=k$ or $a{\left(x\xe2\u02c6\u2019h\right)}^{2}=k$, 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.

#### 10.4 Solve Applications Modeled by Quadratic Equations

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

**Pythagorean Theorem**In any right triangle, where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypothenuse, ${a}^{2}+{b}^{2}={c}^{2}$

**Projectile motion**The height in feet, $h$, of an object shot upwards into the air with initial velocity, ${v}_{0}$, after $t$ seconds can be modeled by the formula:

$$h=\mathrm{\xe2\u02c6\u201916}{t}^{2}+{v}_{0}t$$

#### 10.5 Graphing Quadratic Equations in Two Variables

**The graph of every quadratic equation is a parabola.****Parabola Orientation**For the quadratic equation $y=a{x}^{2}+bx+c$, if- $a>0$, the parabola opens upward.
- $a<0$, the parabola opens downward.

**Axis of Symmetry and Vertex of a Parabola**For a parabola with equation $y=a{x}^{2}+bx+c$:- The axis of symmetry of a parabola is the line $x=\xe2\u02c6\u2019\frac{b}{2a}$.
- The vertex is on the axis of symmetry, so its
*x*-coordinate is $\xe2\u02c6\u2019\frac{b}{2a}$. - To find the
*y*-coordinate of the vertex we substitute $x=\xe2\u02c6\u2019\frac{b}{2a}$ into the quadratic equation.

**Find the Intercepts of a Parabola**To find the intercepts of a parabola with equation $y=a{x}^{2}+bx+c$:

$\begin{array}{cccc}\hfill {\text{y}}\mathbf{\text{-intercept}}& & & \hfill {\text{x}}\mathbf{\text{-intercepts}}\\ \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}x=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}y.& & & \hfill \text{Let}\phantom{\rule{0.2em}{0ex}}y=0\phantom{\rule{0.2em}{0ex}}\text{and solve for}\phantom{\rule{0.2em}{0ex}}x.\end{array}$**To Graph a Quadratic Equation in Two Variables**- Step 1. Write the quadratic equation with $y$ on one side.
- Step 2. Determine whether the parabola opens upward or downward.
- Step 3. Find the axis of symmetry.
- Step 4. Find the vertex.
- 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. - Step 7. 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