# Use Graphs to Solve Quadratic Equations

## Key Questions

• Take the whole function and set it equal to zero, then just solve it like you would a normal equation.

A complex number '$\alpha$' is called a solution or root of a quadratic equation $f \left(x\right) = a {x}^{2} + b x + c$
if $f \left(\alpha\right) = a {\alpha}^{2} + b \alpha + c = 0$

#### Explanation:

If you have a function - $f \left(x\right) = a {x}^{2} + b x + c$
and have a complex number - $\alpha$ .

If you substitute the value of $\alpha$ into $f \left(x\right)$ and got the answer 'zero', then $\alpha$ is said to be the solution / root of the quadratic equation.

There are two roots for a quadratic equation .

Example :

Let a quadratic equation be - $f \left(x\right) = {x}^{2} - 8 x + 15$

The roots of it will be 3 and 5 .

as $f \left(3\right) = {3}^{2} - 8 \cdot 3 + 15 = 9 - 24 + 15 = 0$ and

$f \left(5\right) = {5}^{2} - 8 \cdot 5 + 15 = 25 - 40 + 15 = 0$ .

• Let us look at the following example.

Solve $2 {x}^{2} + 5 x - 3 = 0$ for $x$.

(Be sure to move all terms to the left-hand side so that you have zero on the right-hand side.)

Step 1: Go to "Y=" and type in the quadratic function. Step 2: Go to "WINDOW" to set an appropriate window size. Step 3: Go to "CALC", and choose "zero". Step 4: It will say "Left Bound?", so move the cursor to the left of the zero and press "ENTER". Step 5: It will say "Right Bound?", so move the cursor to the right of the zero and press "ENTER". Step 6: It will say "Guess?", then move the cursor as close to the zero as possible and press "ENTER". Step 7: The zero ($X = - 3$) is displayed at the bottom of the screen. Step 8: Repeat Step 3-7 to find the other zero ($X = 0.5$). I hope that this was helpful.

• If you are solving the quadratic equation of the form

$a {x}^{2} + b x + c = 0$,

and the graph of the quadratic function $y = a {x}^{2} + b x + c$ is available, then you can solve the quadratic equation by finding the x-intercepts (or zeros) of the quadratic function.

I hope that this was helpful.