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At the end of the last section (Completing the Square), we derived a general formula for solving quadratic equations. Here is that general formula:

For any quadratic equation ax^2+ bx + c = 0, the solutions for x can be found by using the quadratic formula:

x=(-b+-sqrt(b^2-4ac))/(2a)

The expression under the square root, b^2− 4ac, can tell us how many roots we'll get. (There's no magic here - just a consideration of what the square root of b^2− 4ac is.)

If  b^2− 4ac = 0, then we'll have one root only, x = −b/(2a).

If  b^2− 4ac > 0, then we'll have two roots, one involving the "+" sign and the other involving the "−" sign in the formula.

If  b^2− 4ac < 0, then we'll have no real roots, since you cannot find the square root of a negative number.

The expression b^2 − 4ac is called the discriminant and in some books you will see it written with a Greek upper case Delta, like this Delta = b^2 − 4ac.

### Example 1

Solve 2x^2- 7x - 5 = 0 using the quadratic formula.

By inspection, we can see that: a = 2, b = -7 and c = -5.

Substituting these into the quadratic formula, we get :

x=(-b+-sqrt(b^2-4ac))/(2a)

=(-(-7)+-sqrt((-7)^2-4(2)(-5)))/(2(2))

=(7+-sqrt(49+40))/4

=(7+-sqrt(89))/4

=(7-sqrt89)/4 or (7+sqrt89)/4

=-0.6085 or 4.108

### Example 2

Solve 2x^2= 4x + 3

Firstly, in order to identify a, b and c , we must re-arrange the expression in the proper form ie. all the terms to the left, leaving zero on the right:

2x^2- 4x - 3 = 0

Only then, we can see that:

a = 2, b = -4 and c = -3

Substituting these into the quadratic formula, we get:

x=(-b+-sqrt(b^2-4ac))/(2a)

x=(-(-4)+-sqrt((-4)^2-4(2)(-3)))/(2(2))

=(4+-sqrt(16+24))/4

=(4+-sqrt40)/4

=(4-sqrt40)/4 or (4+sqrt40)/4

So

x=-0.581 or 2.581

### Exercise

Solve 6r^2= 6r + 1 using the quadratic formula.

6r^2=6r+1

Rearranging so it's in quadratic form:

6r^2-6r-1=0

Using the Quadratic Equation for the variable r:

r=(-b+-sqrt(b^2-4ac))/(2a)

=(-(-6)+-sqrt((-6)^2-4(6)(-1)))/(2(6))

=(6+-sqrt(36+24))/12

=(6+-sqrt60)/12

=(6-2sqrt15)/12 or (6+2sqrt15)/12

=(3-sqrt15)/6 or (3+sqrt15)/6

So

r=-0.145 or 1.145