4.

Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth.

2a2 − 46a + 252 = 0

18, 28

9, 14

–9, –14

–18, 28

1 answer

To solve the quadratic equation \(2a^2 - 46a + 252 = 0\) using the quadratic formula, we first identify the coefficients \(a\), \(b\), and \(c\) from the general form of a quadratic equation \(ax^2 + bx + c = 0\).

For the equation \(2a^2 - 46a + 252 = 0\):
- \(a = 2\)
- \(b = -46\)
- \(c = 252\)

The quadratic formula is given by:

\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substitute the values we identified into the formula:

\[ a = \frac{-(-46) \pm \sqrt{(-46)^2 - 4 \cdot 2 \cdot 252}}{2 \cdot 2} \]

Simplify inside the square root first:

\[ (-46)^2 = 2116 \]

\[ 4 \cdot 2 \cdot 252 = 2016 \]

So, the equation under the square root becomes:

\[ 2116 - 2016 = 100 \]

Now substitute back into the quadratic formula:

\[ a = \frac{46 \pm \sqrt{100}}{4} \]

Since \(\sqrt{100} = 10\), we have:

\[ a = \frac{46 \pm 10}{4} \]

This gives us two possible solutions:

\[ a = \frac{46 + 10}{4} \]

\[ a = \frac{46 - 10}{4} \]

Calculate each solution:

\[ a = \frac{56}{4} = 14 \]

\[ a = \frac{36}{4} = 9 \]

So the solutions to the quadratic equation \(2a^2 - 46a + 252 = 0\) are \(a = 9\) and \(a = 14\).

Thus, the correct answer is:

\[ \boxed{9, 14} \]