Question

x² - 19x + 80 = -8
x² - 5x = -4x
3x² - 147 = 0
9x² + 12x + 4 = 0
x² - 2x + 3 = 0
x² + 6x + 9 = 25

Explanation:

Let's solve each quadratic equation one by one.

1. Equation: \(x^2 - 19x + 80 = -8\)

Step 1: Rewrite in standard form

Add 8 to both sides: \(x^2 - 19x + 88 = 0\)

Step 2: Factor the quadratic

We need two numbers that multiply to 88 and add to -19. The numbers are -8 and -11.
So, \((x - 8)(x - 11) = 0\)

Step 3: Solve for \(x\)

Set each factor equal to zero:
\(x - 8 = 0 \implies x = 8\)
\(x - 11 = 0 \implies x = 11\)

2. Equation: \(x^2 - 5x = -4x\)

Step 1: Rewrite in standard form

Add \(4x\) to both sides: \(x^2 - x = 0\)

Step 2: Factor the quadratic

Factor out \(x\): \(x(x - 1) = 0\)

Step 3: Solve for \(x\)

Set each factor equal to zero:
\(x = 0\)
\(x - 1 = 0 \implies x = 1\)

3. Equation: \(3x^2 - 147 = 0\)

Step 1: Isolate the \(x^2\) term

Add 147 to both sides: \(3x^2 = 147\)
Divide both sides by 3: \(x^2 = 49\)

Step 2: Take the square root

\(x = \pm\sqrt{49} = \pm 7\)

4. Equation: \(9x^2 + 12x + 4 = 0\)

Step 1: Recognize the perfect square trinomial

The quadratic is a perfect square: \((3x + 2)^2 = 0\)

Step 2: Solve for \(x\)

Set \(3x + 2 = 0\)
\(3x = -2 \implies x = -\frac{2}{3}\) (double root)

5. Equation: \(x^2 - 2x + 3 = 0\)

Step 1: Use the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for \(ax^2 + bx + c = 0\)
Here, \(a = 1\), \(b = -2\), \(c = 3\)

Step 2: Calculate the discriminant

\(b^2 - 4ac = (-2)^2 - 4(1)(3) = 4 - 12 = -8\)

Step 3: Solve for \(x\)

Since the discriminant is negative, the solutions are complex:
\(x = \frac{2 \pm \sqrt{-8}}{2} = \frac{2 \pm 2i\sqrt{2}}{2} = 1 \pm i\sqrt{2}\)

6. Equation: \(x^2 + 6x + 9 = 25\)

Step 1: Rewrite in standard form

Subtract 25 from both sides: \(x^2 + 6x - 16 = 0\)

Step 2: Factor the quadratic

We need two numbers that multiply to -16 and add to 6. The numbers are 8 and -2.
So, \((x + 8)(x - 2) = 0\)

Step 3: Solve for \(x\)

Set each factor equal to zero:
\(x + 8 = 0 \implies x = -8\)
\(x - 2 = 0 \implies x = 2\)

Answer:

s:

1. \(x^2 - 19x + 80 = -8\): \(x = 8\) or \(x = 11\)
2. \(x^2 - 5x = -4x\): \(x = 0\) or \(x = 1\)
3. \(3x^2 - 147 = 0\): \(x = 7\) or \(x = -7\)
4. \(9x^2 + 12x + 4 = 0\): \(x = -\frac{2}{3}\) (double root)
5. \(x^2 - 2x + 3 = 0\): \(x = 1 + i\sqrt{2}\) or \(x = 1 - i\sqrt{2}\)
6. \(x^2 + 6x + 9 = 25\): \(x = -8\) or \(x = 2\)