Question

solve each equation by using the quadratic formula.
$x^2 - 14x + 53 = 0$
$\bigcirc\\ 7\pm 2i$
$\bigcirc\\ 7\pm 4i$
$\bigcirc\\ 7, 2i$
$\bigcirc\\ 5, 9$

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b=-14\), \(c = 53\).

Step2: Apply Quadratic Formula

Quadratic Formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x=\frac{-(-14)\pm\sqrt{(-14)^2 - 4(1)(53)}}{2(1)}\)

Step3: Simplify discriminant

Calculate \(b^2 - 4ac\): \((-14)^2-4(1)(53)=196 - 212=-16\)

Step4: Simplify square root

\(\sqrt{-16}=\sqrt{16}\cdot\sqrt{-1}=4i\) (since \(\sqrt{-1}=i\))

Step5: Solve for x

\(x=\frac{14\pm4i}{2}=\frac{14}{2}\pm\frac{4i}{2}=7\pm2i\)

Answer:

\(7 \pm 2i\) (corresponding to the option "7 ± 2i")