Question

Find the x-intercepts of the graph of the equation. $$y=x^{2}+7 x-2$$

Short answer

The x-intercepts of the graph of the equation \(y=x^{2}+7 x-2\) are \(x = \frac{-7 + \sqrt{57}}{2}\) and \(x = \frac{-7 - \sqrt{57}}{2}\)

Step-by-step solution

Setting the Equation to Zero

To find the x-intercepts, set the equation equal to zero: \(0 = x^{2} +7x -2\). This is because x-intercepts are the x-coordinates where the graph of a function intersects the x-axis, at which point \(y = 0\).

Identify the coefficients of the quadratic equation

A general quadratic equation is given as \(ax^2 + bx + c = 0\). In this equation, \(a = 1\), \(b = 7\) and \(c = -2\).

Apply the Quadratic Formula

Applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), substitute the identified values from Step 2. This gives \(x = \frac{-7 \pm \sqrt{7^2 - 4*1*(-2)}}{2*1}\). Simplify under the square root sign to obtain \(x = \frac{-7 \pm \sqrt{49 + 8}}{2}\).

Final Calculation

Perform the final calculations to get the two solutions for \(x\): \(x = \frac{-7 \pm \sqrt{57}}{2}\). Therefore, the 2 solutions or x intercepts of the equation are \(x = \frac{-7 + \sqrt{57}}{2}\) and \(x = \frac{-7 - \sqrt{57}}{2}\).