1.x²+5x-14=0 2.x²-4x+12=0 solve quadratic equation using the quadratic formula

To solve the quadratic equations using the quadratic formula, we'll follow the standard formula:

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

For each equation, we'll identify the coefficients \(a\), \(b\), and \(c\) and then apply the formula.

### 1. \(x^2 + 5x - 14 = 0\)
Here, the coefficients are:
- \(a = 1\)
- \(b = 5\)
- \(c = -14\)

Plugging these values into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-14)}}{2(1)}
\]

\[
x = \frac{-5 \pm \sqrt{25 + 56}}{2}
\]

\[
x = \frac{-5 \pm \sqrt{81}}{2}
\]

\[
x = \frac{-5 \pm 9}{2}
\]

So the two possible solutions are:

\[
x_1 = \frac{-5 + 9}{2} = \frac{4}{2} = 2
\]

\[
x_2 = \frac{-5 - 9}{2} = \frac{-14}{2} = -7
\]

### Solutions:
\[
x = 2 \quad \text{or} \quad x = -7
\]

### 2. \(x^2 - 4x + 12 = 0\)
Here, the coefficients are:
- \(a = 1\)
- \(b = -4\)
- \(c = 12\)

Plugging these values into the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(12)}}{2(1)}
\]

\[
x = \frac{4 \pm \sqrt{16 - 48}}{2}
\]

\[
x = \frac{4 \pm \sqrt{-32}}{2}
\]

Since \(\sqrt{-32}\) is an imaginary number, we simplify:

\[
x = \frac{4 \pm \sqrt{32}i}{2}
\]

\[
x = \frac{4 \pm 4\sqrt{2}i}{2}
\]

\[
x = 2 \pm 2\sqrt{2}i
\]

### Solutions:
\[
x = 2 + 2\sqrt{2}i \quad \text{or} \quad x = 2 - 2\sqrt{2}i
\]

These are complex solutions.

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Would you like further details or explanations on any part of this process?

Here are 5 related questions for further practice:
1. Solve \(2x^2 - 3x + 1 = 0\) using the quadratic formula.
2. Solve \(3x^2 + 6x + 2 = 0\) using the quadratic formula.
3. Solve \(x^2 + 2x + 5 = 0\) using the quadratic formula.
4. Determine the nature of the roots for the equation \(x^2 + 6x + 10 = 0\).
5. Solve \(4x^2 - 4x + 1 = 0\) using the quadratic formula.

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before solving a quadratic equation to know if the roots will be real, repeated, or complex.

Learn how to solve quadratic equations x^2 + 5x - 14 and x^2 - 4x + 12 using the quadratic formula. Understand the steps involved and discover the nature of the roots. Suitable for high school students.

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