# How do you use the discriminant to determine the nature of the solutions given 4x^2 + 12x = 14?

Feb 13, 2017

$368 > 0$ and not a perfect square

Therefore, the roots will be real and unequal

#### Explanation:

When you are solving a quadratic equation, it is very useful to know what sort of answer you will get. This can often help in determining which method to use - for example whether to look for factors or to use the quadratic formula.

A quadratic equation is written in the form $a {x}^{2} + b x + c = 0$
Always change to this form first

The discriminant is $\Delta = {b}^{2} - 4 a c$
The solutions to an equation are called the 'roots' and are referred to as $\alpha \mathmr{and} \beta$

The value of $\Delta$ tells us about the nature of the roots.

If $\Delta > 0 \Rightarrow$ the roots are real and unequal (2 distinct roots)

If $\Delta > 0 \text{ and a prefect square} \Rightarrow$ the roots are real, unequal and$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \ldots \ldots \ldots \ldots . .}$ rational

If $\Delta = 0 \Rightarrow$ the roots are real and equal (1 root)

If $\Delta < 0 \Rightarrow$ the roots are imaginary and unequal

Note that if $a \text{ or } b$ are irrational, the roots will be irrational.

$4 {x}^{2} + 12 x = 14 \text{ "rArr} 4 {x}^{2} + 12 x - 14 = 0$

$\Delta = {b}^{2} - 4 a c$

$\Delta = {\left(12\right)}^{2} - 4 \left(4\right) \left(- 14\right) = 368$

$368 > 0$ and not a perfect square

Therefore, the roots will be real and unequal