Solving Quadratic Equations
A quadratic equation is an equation that could be written as
ax 2 + bx + c = 0
when a 0.
There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, & completing the square.
FactoringTo solve a quadratic equation by factoring,
Put all terms on one side of the equal sign, leaving zero on the other side.
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Factor.
Set each factor equal khổng lồ zero.
Solve each of these equations.
Cheông chồng by inserting your answer in the original equation.
Example 1Solve sầu x 2 – 6 x = 16.
Following the steps,
x 2 – 6 x = 16 becomes x 2 – 6 x – 16 = 0
Factor.
( x – 8)( x + 2) = 0
Setting each factor to zero,
Then khổng lồ check,
Both values, 8 và –2, are solutions lớn the original equation.
Example 2Solve sầu y 2 = – 6 y – 5.
Setting all terms equal khổng lồ zero,
y 2 + 6 y + 5 = 0
Factor.
( y + 5)( y + 1) = 0
Setting each factor khổng lồ 0,
To check, y 2 = –6 y – 5
A quadratic with a term missing is called an incomplete quadratic (as long as the ax 2 term isn"t missing).
Example 3Solve x 2 – 16 = 0.
Factor.
To check, x 2 – 16 = 0
Solve x 2 + 6 x = 0.
Factor.
To kiểm tra, x 2 + 6 x = 0
Solve sầu 2 x 2 + 2 x – 1 = x 2 + 6 x – 5.
First, simplify by putting all terms on one side and combining lượt thích terms.
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Now, factor.
To kiểm tra, 2 x 2 + 2 x – 1 = x 2 + 6 x – 5
Many quadratic equations cannot be solved by factoring. This is generally true when the roots, or answers, are not rational numbers. A second method of solving quadratic equations involves the use of the following formula:
a, b, and c are taken from the quadratic equation written in its general khung of
ax 2 + bx + c = 0
where a is the numeral that goes in front of x 2, b is the numeral that goes in front of x, and c is the numeral with no variable next to it (a.k.a., “the constant”).
When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b 2 – 4 ac. A quadratic equation with real numbers as coefficients can have sầu the following:
Two different real roots if the discriminant b 2 – 4 ac is a positive number.
One real root if the discriminant b 2 – 4 ac is equal to 0.
No real root if the discriminant b 2 – 4 ac is a negative number.
Example 6Solve sầu for x: x 2 – 5 x = –6.
Setting all terms equal khổng lồ 0,
x 2 – 5 x + 6 = 0
Then substitute 1 (which is understood lớn be in front of the x 2), –5, and 6 for a, b, & c, respectively, in the quadratic formula and simplify.
Because the discriminant b 2 – 4 ac is positive, you get two different real roots.
Exampleproduces rational roots. In Example, the quadratic formula is used to lớn solve an equation whose roots are not rational.
Example 7Solve for y: y 2 = –2y + 2.
Setting all terms equal lớn 0,
y 2 + 2 y – 2 = 0
Then substitute 1, 2, & –2 for a, b, & c, respectively, in the quadratic formula và simplify.
Note that the two roots are irrational.
Example 8Solve sầu for x: x 2 + 2 x + 1 = 0.
Substituting in the quadratic formula,
Since the discriminant b 2 – 4 ac is 0, the equation has one root.
The quadratic formula can also be used to solve sầu quadratic equations whose roots are imaginary numbers, that is, they have no solution in the real number system.
Example 9Solve sầu for x: x( x + 2) + 2 = 0, or x 2 + 2 x + 2 = 0.
Substituting in the quadratic formula,
Since the discriminant b 2 – 4 ac is negative sầu, this equation has no solution in the real number system.
But if you were lớn express the solution using imaginary numbers, the solutions would be
A third method of solving quadratic equations that works with both real và imaginary roots is called completing the square.
Put the equation into lớn the khung ax 2 + bx = – c.
Make sure that a = 1 (if a ≠ 1, multiply through the equation by
Using the value of b from this new equation, add
Find the square root of both sides of the equation.
Solve the resulting equation.
Example 10Solve sầu for x: x 2 – 6 x + 5 = 0.
Arrange in the size of
Because a = 1, add
Take the square root of both sides.
x – 3 = ±2
Solve.
Solve for y: y 2+ 2 y – 4 = 0.
Arrange in the khung of
Because a = 1, add
Take the square root of both sides.
Solve.
Solve sầu for x: 2 x 2 + 3 x + 2 = 0.
Arrange in the size of
Because a ≠ 1, multiply through the equation by
Add
Take the square root of both sides.
There is no solution in the real number system. It may interest you khổng lồ know that the completing the square process for solving quadratic equations was used on the equation ax 2 + bx + c = 0 to lớn derive the quadratic formula.
