Question

solve each equation using the quadratic formula. write answers in simplest radical form.
1.) $2x^{2}+5x + 4 = 0$
2.) $4x^{2}+8x = 96$
3.) $2x^{2}-7x - 13 = -10$
4.) $x^{2}-42 = -2x$
5.) $x^{2}-4x + 4 = 0$
6.) $2x^{2}+3x = 20$

Explanation:

Problem 1: \(2x^2 + 5x + 4 = 0\)

Step1: Identify \(a\), \(b\), \(c\)

For \(ax^2 + bx + c = 0\), here \(a = 2\), \(b = 5\), \(c = 4\).

Step2: Quadratic Formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)

Calculate discriminant \(D = b^2 - 4ac = 5^2 - 4\times2\times4 = 25 - 32 = -7\).
Since \(D<0\), solutions are complex: \(x=\frac{-5\pm\sqrt{-7}}{4}=\frac{-5\pm i\sqrt{7}}{4}\).

Step1: Rewrite in standard form

\(4x^2 + 8x - 96 = 0\), divide by 4: \(x^2 + 2x - 24 = 0\). So \(a = 1\), \(b = 2\), \(c = -24\).

Step2: Apply Quadratic Formula

Discriminant \(D = 2^2 - 4\times1\times(-24)=4 + 96 = 100\).
\(x=\frac{-2\pm\sqrt{100}}{2}=\frac{-2\pm10}{2}\).

Step3: Solve for \(x\)

\(x_1=\frac{-2 + 10}{2}=4\), \(x_2=\frac{-2 - 10}{2}=-6\).

Step1: Rewrite in standard form

\(2x^2 - 7x - 3 = 0\). So \(a = 2\), \(b = -7\), \(c = -3\).

Step2: Quadratic Formula

Discriminant \(D = (-7)^2 - 4\times2\times(-3)=49 + 24 = 73\).
\(x=\frac{7\pm\sqrt{73}}{4}\).

Answer:

\(x = \frac{-5 + i\sqrt{7}}{4}\) or \(x = \frac{-5 - i\sqrt{7}}{4}\)

Problem 2: \(4x^2 + 8x = 96\)