Question

1. $2x^2 + 4x - 57 = 87$

Explanation:

Step1: Rearrange the equation

First, we want to set the quadratic equation to standard form \(ax^2 + bx + c = 0\). So we subtract 87 from both sides of the equation \(2x^2 + 4x - 57 = 87\).
\[
2x^2 + 4x - 57 - 87 = 0
\]
\[
2x^2 + 4x - 144 = 0
\]

Step2: Simplify the equation

We can divide the entire equation by 2 to simplify it.
\[
\frac{2x^2}{2}+\frac{4x}{2}-\frac{144}{2}=0
\]
\[
x^2 + 2x - 72 = 0
\]

Step3: Use the quadratic formula

For a quadratic equation \(ax^2+bx + c = 0\), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 2\), and \(c=-72\).
First, calculate the discriminant \(\Delta=b^2 - 4ac=(2)^2-4\times1\times(-72)=4 + 288 = 292\).
Then, find the square root of the discriminant \(\sqrt{292}=\sqrt{4\times73}=2\sqrt{73}\).
Now, substitute into the quadratic formula:
\[
x=\frac{-2\pm2\sqrt{73}}{2\times1}=\frac{-2\pm2\sqrt{73}}{2}=- 1\pm\sqrt{73}
\]
Calculating the numerical values: \(\sqrt{73}\approx8.544\), so \(x=-1 + 8.544\approx7.544\) or \(x=-1-8.544\approx - 9.544\)

Answer:

The solutions are \(x=-1+\sqrt{73}\approx7.54\) and \(x=-1 - \sqrt{73}\approx - 9.54\) (or in exact form \(x=-1\pm\sqrt{73}\))