Question

Solve the given equation by using the quadratic formula.

7a. x^2+3x-2=0

7b. 7x^2-2x=-5

Answer 1

The quadratic formula is $x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

7a. In this equation $a = 1$, $b = 3$, $c = -2$. Put it into the formula:

$$x_{1,2} = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{-3 \pm \sqrt{17}}{2}$$

Answer: $x_{1,2} = \frac{-3 \pm \sqrt{17}}{2}$

7b. In this equation $a = 7$, $b = -2$, $c = 5$. Put it into the formula:

$$x_{1,2} = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 7 \cdot 5}}{2 \cdot 7} = \frac{2 \pm \sqrt{4 - 140}}{14}$$

We see that discriminant $D = b^{2} - 4ac < 0$, then there are not real solutions.

Complex solution: $x_{1,2} = \frac{2 \pm \sqrt{4 - 140}}{14} = \frac{1 \pm i\sqrt{34}}{7}$