Q20. - Algebra 1

Question

Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.

$x^2-10x=-21$

Step-by-Step Solution

Answer

The roots of the equation $x^2-10x=-21$ are $\mathit{x}\mathbf{=}\mathbf{3}$ and $\mathit{x}\mathbf{=}\mathbf{7}$.  

Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $f\left(x\right)=a{x}^2+bx+c$, where, $a\ne 0$ is called the standard form of the quadratic function.

Step 2. Rewrite the equation x 2 − 10 x = − 21 in the form f x = a x 2 + b x + c .

Write the equation $x^2-10x=-21$ in standard form.

This equation is already in standard form.

Write the equation $x^2-10x+21=0$ in the form $f\left(x\right)=a{x}^2+bx+c$.

$f\left(x\right)=x^2-10x+21$

Step 3. Plot the graph of the function f x = x 2 − 10 x + 21 .

The graph of the function $f\left(x\right)=x^2-10x+21$ is shown below.

Step 4. Solve the equation x 2 − 10 x = − 21 from the graph of the function f x = x 2 − 10 x + 21 .

Observe the graph of the function $f\left(x\right)=x^2-10x+21$.

The graph intersects the  $x$-axis at the points $x=3$ and $x=7$.  

Therefore, the roots of the equation $x^2-10x=-21$ are $x=3$ and $x=7$.