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# Solving Quadratic Equations: The Methods

10

Quadratic Equations

Algebra

###### Quadratic equations are equations in the form:

###### … where a, b, and c are constants. The solutions of quadratic equations are the values of x that make the equation true.

###### In this guide, we will show you how to solve quadratic equations using a number of methods...

###### 1. Factorising

###### 2. Difference of Two Squares

###### 3. Quadratics in the form ax^2 + bx = 0

###### 4. Factorising when a is greater than 1

###### 5. Completing the square

###### 6. Using the Quadratic Equation

###### 7. The Discriminant

**1. Factorising**

###### When a quadratic expression is in the form x^2 + bx + c find two numbers that add to give b and multiply to give c.

##### Example One:

###### Solve the following quadratic equation by factorising.

###### This equation is already in standard form.

###### If the quadratic equation is factorable, you can factorise it into two binomials in the form (x + r)(x + s) = 0, where r and s are constants.

###### We can factorise this equation by finding two numbers that add up to –1 and multiply to get –12.

###### A good systematic way is to look at factors of 12:

###### 1 & 12

###### 2 & 6

###### 3 & 4

###### The only pair that will work is 3 & 4:

###### 3 – 4 = –1

###### If each bracket is set to equal 0, we can get solutions for x:

###### x – 4 = 0, x = 4

###### x + 3 = 0, x = –3

###### x = 4 and x = –3 are our solutions

##### Example Two:

###### Solve the following quadratic equation by factorising..

###### Looking at factors of 8:

###### 1 & 8

###### 2 & 4 (This looks good!)

###### The solutions are: x = 2 and x = –4

**2. Difference of Two Squares**

###### Expressions in the form

###### can be factorised to give:

###### Examples

**3. ****Quadratics in the form ax^2 + bx = 0**

###### Pretty easy, uh...

**4. Factorising when a is greater than 1**

###### When a quadratic is in the form

###### Multiply a by c = ac

###### Find two numbers that will add to give b and multiply to give ac.

###### Re-write the quadratic and replace bx with the two numbers.

###### Factorise in pairs.

###### Exercise

###### Factorise the following quadratic

**5. Completing the square**

###### Divide all terms by a if a > 1

###### Move the number term to the other side of the equation.

###### Complete the square on the left side of the equation - balance the c-squared value in the bracket by adding the same value to the right side of the equation.

###### Take the square root on both sides of the equation.

###### Move the number that remains on the left side of the equation to the right to find the values for x.

###### You could solve without changing the fractions into decimals

###### Key moment here;

###### Dividing the 0.8 by 2 as you complete the square then adding the square of the bracketed number to the other side of the equation.

###### Here's another example...

###### One more for luck...

**6. Using the Quadratic Equation**

###### The method of last resort....

###### If the quadratic equation is not factorable, you can use the quadratic formula - it works every time:

###### where a, b, and c are the constants from your quadratic equation. This formula gives you two solutions for x.

###### Once you have found your solutions for x, simplify them if possible. If your solutions are irrational numbers, you can leave them in that form or approximate them to a decimal.

###### Example:

###### We used this expression earlier

###### Using the quadratic formula...

###### Putting the values in:

**7. The Discriminant**

###### The discriminant of a quadratic formula is a term that appears in the quadratic formula and is given by:

###### The discriminant tells us information about the nature and number of solutions (or roots) of the quadratic equation.

###### If the discriminant is...

###### positive = two roots, The equation intersects the x-axis at two points.

###### zero = one real root & equation touches the x-axis at exactly one point.

###### negative = two complex conjugate roots, equation does not intersect the x-axis.

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