Solve the following quadratic equation for all values of xx in simplest form.

Question

4, left parenthesis, x, squared, plus, 5, right parenthesis, minus, 17, equals, 12
4(x
2
+5)−17=
12

Answer 1

To solve the given quadratic equation, we start with:

$$4(x^2+5)-17=12.$$

First, let's simplify the left side of the equation:

1. Distribute the 4: $$4x^2+20-17=12.$$

2. Combine the constant terms (20 and -17): $$4x^2+3=12.$$

Next, we need to isolate the $x^2$ term:

3. Subtract 3 from both sides: $$4x^2=12-3,$$ which simplifies to: $$4x^2=9.$$

4. Now, divide both sides by 4: $$x^2=\frac{9}{4}.$$

5. To find $x$, take the square root of both sides: $$x=\pm \sqrt{\frac{9}{4}}.$$

The square root of a fraction can be simplified: $$x=\pm \frac{\sqrt{9}}{\sqrt{4}}=\pm \frac{3}{2}.$$

Thus, the solutions for $x$ are: $$x=\frac{3}{2}\text{and}x=-\frac{3}{2}.$$

In conclusion, the values of $x$ in simplest form are: $$\boxed{{\displaystyle \frac{3}{2}\text{ and }-\frac{3}{2}}}.$$