Question

Represent the solution graphically. Check the solution algebraically. $$ x^{2}+4 x=21 $$

Short answer

To solve the equation \(x^{2}+4x=21\), you need to first rewrite it into the standard form as \(x^{2}+4x-21=0\). Afterwards, use the quadratic formula to find the roots \(x_1\) and \(x_2\). Plot the function \(f(x)=x^{2}+4x-21\) to represent the solution graphically. \(x_1\) and \(x_2\) are where the curve intersects the x-axis. Finally, check the correctness of our solution by substituting \(x_1\) and \(x_2\) into our initial equation. If both sides match, then the solution is verified.

Step-by-step solution

Rewrite the Equation

Rewrite the given equation \(x^{2}+4x=21\) to standard form, which yields \(x^{2}+4x-21=0\)

Solve the Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt {b^{2}-4ac}}{2a}\) to solve for \(x\). Inputs for the formula are \(a=1\), \(b=4\), and \(c=-21\). Calculating that yields two solutions \(x_1\) and \(x_2\).

Graph the Function and Solutions

Plot the function \(f(x)=x^{2}+4x-21\). The solutions obtained will be the zeros of the function. The solutions \(x_1\) and \(x_2\) graphically represents the points where the curve intersects the x-axis.

Check the Solution

Substitute the solutions \(x_1\) and \(x_2\) into the original equation to check the solution. If both sides of the equation are equal, then the solution is correct.