Solve quadratic equations by using the quadratic formula
How to solve quadratic equations using the quadratic formula
To identify a quadratic equation a symbol is used, that symbol is ‘^{2}’.
Here's an example of a quadratic equation:
2x^{2}+5x+8=0
The squared symbol, highlighted in red, lets us know that the equation is a quadratic equation. Quadratic comes from ‘quad’ meaning square, that’s another way to identify a quadratic equation.
The standard form of a quadratic equation is:
ax^{2}+bx+c=0
The letters a, b and c are coefficients (you will always know the value of a,b and c). a, b and c can be any value, except ‘a’ can’t be 0.
The letter x is unknown to us, and this is the value we are trying to calculate. For example,
2x^{2}+5x+8=0
a = 2, b = 5 and c = 8
There are three methods of how to solve quadratic equations:
1. You can factor the quadratic
2. You can use the method of completing the square
3. You can use the quadratic formula
Method 3, using the formula is what we are going to use today, the formula that is always used to solve quadratic equations is:
x=b+√b^{2}4ac
2a
At first glance it looks daunting, but if you break it up then it isn’t too bad. All we do is substitute a, b and c for values and then just enter it into a calculator.
Now let's solve 5x^{2}+6x+1=0 we just have to follow these simple steps:
1 Write out the quadratic formula
x=b+√b^{2}4ac
2a
2 Write out the quadratic equation and pick out a, b and c.
5x^{2}+6x+1=0
a = 5, b = 6 and c = 1
3 Substitute your a, b and c values into the formula.
x=6+√6^{2}4(5)(1)
2(5)
4 Simplify the areas you can.
x=6+√3620 = x=6+√16 = x=6+4
10 10 10
5 This part is important, we must do two calculations as there are two possible answers for x
.x=64 and we must do x=6+4
10 10
6 Enter those into your calculator. Your answer will be:
x = 1 or 0.2
Examples
Solve 3x^{2}+6x+9=0 using the quadratic formula.
3x^{2}+6x+2=0
a=3, b=6 and c=2
x=b+√b^{2}4ac
2a
x=6+√6^{2}4(3)(2)
2(3)
x=6+√3624
x=6+3.46 (2d.p)
6
x=6+3.46 = 0.42 (2d.p) or x=63.46 = 1.58 (2d.p)
6 6
Solve 4x^{2}+3x+5=0 using the quadratic formula.
4x^{2}+10x+5=0
a=4, b=10 and c=5
x=b+√b^{2}4ac
2a
x=3+√10^{2}4(4)(5)
2(4)
x=3+√10080
8
x=3+4.47 (2d.p)
8
x=3+4.47=0.18 (2d.p) or x=34.47=0.93 (2d.p)
8 8
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