Question

Represent the solution graphically. Check the solution algebraically. $$ x^{2}-x=6 $$

Short answer

The solutions of the quadratic equation \(x^{2}-x-6 = 0\) are \(x= -2\) and \(x =3\). These are verified by substituting back into the original equation and through graphical representation.

Step-by-step solution

Rearrange the equation

Rearrange the equation to have all terms on one side: \( x^{2}-x-6 = 0 \)

Solve using the quadratic formula

Use the quadratic formula which is given by \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). In this case \(a=1\), \(b=-1\), and \(c=-6\). So, \(x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4*1*-6}}{2*1}\) which simplifies to \(x = \frac{1 \pm \sqrt{1 + 24}}{2}\) and gives two solutions \(x = 3\) and \(x = -2\).

Check solutions

Check the solutions by substituting back into the original equation. For \(x=3\), substition gives \(3^{2} - 3 = 6\). For \(x=-2\), substitution gives \((-2)^{2}-(-2)=6\). Both solutions verify.

Graph the quadratic function

On a graph, plot the function \(f(x) = x^{2}-x-6\). The x-intercepts are \(x= -2\) and \(x =3\), which confirms the algebraic solutions.