Solve $x^2 = 49$.

How we read your problem

A quadratic equation in x: solve with the quadratic formula.

Step-by-step solution

1. 1

   Start with the equation

   $${ x}^{2} = 49$$
2. 2

   Move every term to one side to get standard form

   $$x^{2} - 49 = 0$$
3. 3

   Compute the discriminant

   $$\Delta = b^2 - 4ac = 0^2 - 4 \cdot 1 \cdot (-49) = 196$$
4. 4

   Δ > 0, so there are two distinct real roots

5. 5

   Apply the quadratic formula

   $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{0 \pm \sqrt{196}}{2} = \frac{0 \pm 14}{2}$$
6. 6

   Simplify each root

   $$x_1 = \frac{0 - 14}{2} = -7,\quad x_2 = \frac{0 + 14}{2} = 7$$
7. 7

   Factored form

   $$x^2 - 49 = (x + 7)(x - 7)$$

Formulas used

• Quadratic formula

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

• Discriminant

  $$\Delta = b^2 - 4ac$$

• Vertex

  $$\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)$$

The quadratic formula gives every solution of ax² + bx + c = 0.

Verification

• x = -7: left side = 49, right side = 49 ✓
• x = 7: left side = 49, right side = 49 ✓