Solve the equation $$ x^{2}=5 x+14 $$

Question

UpStudy AI · Accepted Answer

To solve the quadratic equation:

$$
x^{2} = 5x + 14
$$

follow these steps:

### 1. Rewrite the Equation in Standard Form

First, bring all terms to one side of the equation to set it to zero:

$$
x^{2} - 5x - 14 = 0
$$

This is the standard form of a quadratic equation:

$$
ax^{2} + bx + c = 0
$$

where:
- $$ a = 1 $$
- $$ b = -5 $$
- $$ c = -14 $$

### 2. Use the Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation:

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

Plugging in the values:

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

$$
x = \frac{5 \pm \sqrt{25 + 56}}{2}
$$

$$
x = \frac{5 \pm \sqrt{81}}{2}
$$

$$
x = \frac{5 \pm 9}{2}
$$

### 3. Solve for the Two Possible Values of $$ x $$

1. **First Solution ($$+$$ sign):**
   $$
   x = \frac{5 + 9}{2} = \frac{14}{2} = 7
   $$

2. **Second Solution ($$-$$ sign):**
   $$
   x = \frac{5 - 9}{2} = \frac{-4}{2} = -2
   $$

### 4. Conclusion

The solutions to the equation $$ x^{2} = 5x + 14 $$ are:

$$
x = 7 \quad \text{and} \quad x = -2
$$

### Verification

To ensure the solutions are correct, substitute them back into the original equation.

1. **For $$ x = 7 $$:**
   $$
   7^{2} = 5(7) + 14 \implies 49 = 35 + 14 \implies 49 = 49 \quad \text{✓}
   $$

2. **For $$ x = -2 $$:**
   $$
   (-2)^{2} = 5(-2) + 14 \implies 4 = -10 + 14 \implies 4 = 4 \quad \text{✓}
   $$

Both solutions satisfy the original equation.
The solutions to the equation $$ x^{2} = 5x + 14 $$ are $$ x = 7 $$ and $$ x = -2 $$.

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