# Solutions Using the Discriminant

Algebra II - Quadratic Equations - Discriminant

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

$\Delta = {b}^{2} - 4 a c$ for a quadratic $a {x}^{2} + b x + c = 0$

#### Explanation:

The discriminant indicated normally by $\Delta$, is a part of the quadratic formula used to solve second degree equations.
Given a second degree equation in the general form:
$a {x}^{2} + b x + c = 0$
the discriminant is:
$\Delta = {b}^{2} - 4 a c$

The discriminant can be used to characterize the solutions of the equation as:
1) $\Delta > 0$ two separate real solutions;
2) $\Delta = 0$ two coincident real solutions (or one repeated root);
3) $\Delta < 0$ no real solutions.

For example:
${x}^{2} - x - 2 = 0$
Where: $a = 1$, $b = - 1$ and $c = - 2$
So:
$\Delta = {b}^{2} - 4 a c = 1 + 4 \cdot 2 = 9 > 0$, giving $2$ real distinct solutions.

The discriminant can also come in handy when attempting to factorize quadratics. If $\Delta$ is a square number, then the quadratic will factorize, (since the square root in the quadratic formula will be rational). If it is not a square number, then the quadratic will not factorize. This can save you spending ages trying to factorize when it won't work. Instead, solve by completing the square or using the formula.

I hope that helps!

• The solutions to an equation can be determined by the determinants in this manner; where the discriminant intersects with the x-axis provides a set of solutions, taking the values of x at the point. For discriminant,d. if d>0; solutions =2, d<0; no real solutions, d=0; solution=1

• The discriminant is used to determine if the graph (parabola) crosses the x-axis.

In the quadratic formula, the discriminant, is the ${b}^{2} - 4 a c$. B, A, and C are coefficients from the given equation.

1. If this number is positive, then there are 2 places where the parabola crosses the x-axis.
2. If this number is negative, then the parabola does not cross the x-axis at all.
3. If this number is zero, then the parabola only crosses the x-axis once, at its vertex.

Let's do an example and find the discriminant of $2 {x}^{2} - 1 x - 3 = 0$.

From the equation:
$A = 2$, the number in front of ${x}^{2}$
$B = - 1$, the number in front of $x$
$C = - 3$, the number by itself.

Substitute everything into ${b}^{2} - 4 a c$ and you get this

${\left(- 1\right)}^{2} - 4 \left(2\right) \left(- 3\right)$

$1 - 8 \left(- 3\right)$
$1 + 24$
$25$

Remember, you don't care about the number, only if it is positive, negative or zero.

Since this is positive, the graph (parabola) crosses the x-axis TWICE!

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