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

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

**10.3 Solve Quadratic Equations Using the Quadratic Formula**

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

**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*2−4*ac*, to Determine the Number of Solutions of a Quadratic Equation

For a quadratic equation of the form*ax*2+*bx*+*c*=0,*a*≠0,

- if
*b*2−4*ac*>0, the equation has 2 solutions. - if
*b*2−4*ac*=0, the equation has 1 solution. - if
*b*2−4*ac*<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 *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.

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

**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:.

**The graph of every quadratic equation is a parabola.****Parabola Orientation**For the quadratic equation*y*=*ax*^{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*=*ax*^{2}+*bx*+*c*:- The axis of symmetry of a parabola is the line
*x*=−*b*/2*a*. - The vertex is on the axis of symmetry, so its
*x*-coordinate is −*b*/2*a*. - To find the
*y*-coordinate of the vertex we substitute*x*=−*b*/2*a*into the quadratic equation.

- The axis of symmetry of a parabola is the line

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

y**-intercept**x**-intercepts**

Let*x*=0 and solve for*y*. Let*y*=0and solve for*x*.

**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

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