Revision:Quadratic functions, completing the Square, the discriminant and their graphs

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Revision:Quadratic functions, completing the Square, the discriminant and their graphs

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics Revision Notes > Quadratic functions, completing the Square, the discriminant and their graphs

Quadratic functions are polynomials of the 2nd degree. Polynomials being the term to describe the sum of different powers and constants multiplied by a coefficient.

1 Quadratic Functions
2 Factorising
3 Completing the Square
4 Quadratic Formula

Quadratic Functions

The general equation for a quadratic is y = ax^2 + bx + c and $a \ne 0$ .

Quadratic equations have two solutions. To Solve quadratic equations you to first put it in the form ax^2 + bx + c = 0

There are three ways to solve quadratics you need to be familiar with.

Factorising
Completing the Square
Quadratic Equation

Factorising

In the form ax+bx+c=0 when a = 1 this consists of finding a pair of numbers which multiply together to make the c term and add together to make the b term. For example:

x^2+6x+8=0
2 and 4 multiply to make 8 and add to make 8. (x+2)(x+4)=0

To solve one of the brackets much each 0, meaning x + 2 = 0
x = -2
x + 4 = 0
x = -4
x = -2 or -4

Completing the Square

Start by putting the equation in the form ax^2+bx+c=0 . To complete the square we must put the equation in the form a(x+p)^2+q=0.

Take the example 2x^2+12x+9=0
First, factorize out the a term (2) from the first two terms: 2[x^2+6x]+9=0

We then divide the b (6) by two, then put it in brackets with x, then square it. We must also make the constant within the outer brackets equal to 0, giving:

2[(x+3)^2-9]+9=0
We then expand and collect like terms, leaving: 2(x+3)^2-9=0 This is in the completed square form, but is not solved.

We bring the constant (-9) over to the other side, then divide by the a term (2), giving: $(x+3)^2=\frac{9}{2}$
We then square root each side: $x+3=\pm\sqrt{\frac{9}{2}}$

To finish off, we leave x on its own, giving:
$x=-3\pm\sqrt{\frac{9}{2}$

Quadratic Formula

The quadratic formula is $x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ . This is incidentally derived from the method of completing the square.

This formula can be used when the equation is equal to 0 and in the form ax^2 + bx + c = 0 . To find the 2 values of x substitute the values a, b and c into the formula.

$\begin{equation*} a x^2+bx+c &= 0 \\ x^2 + \frac{b}{a}x + \frac{c}{a} &= 0 \\ \left (x+\frac{b}{2a} \right )^2 - \frac{b^2}{4a^2} + \frac{c}{a} &= 0 \\ \left (x+\frac{b}{2a} \right )^2 &= \frac{b^2}{4a^2} - \frac{c}{a} \\ \left (x+\frac{b}{2a} \right )^2 &= \frac{b^2-4ac}{4a^2} \\ x+\frac{b}{2a} &= \frac{\pm \sqrt{b^2-4ac}}{2a} \\ x &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{equation*}$