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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.

Factoring

To 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 1

Solve 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 2

Solve 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 3

Solve x 2 – 16 = 0.

Factor.

*

To check, x 2 – 16 = 0

*

Example 4

Solve x 2 + 6 x = 0.

Factor.

*

To kiểm tra, x 2 + 6 x = 0

*

Example 5

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

*

The quadratic formula

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 6

Solve 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 7

Solve 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 8

Solve 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 9

Solve 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

*
.

Completing the square

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

*
before proceeding).

Using the value of b from this new equation, add

*
lớn both sides of the equation to khung a perfect square on the left side of the equation.

Find the square root of both sides of the equation.

Solve the resulting equation.

Example 10

Solve sầu for x: x 2 – 6 x + 5 = 0.

Arrange in the size of

*

Because a = 1, add

*
, or 9, to lớn both sides to complete the square.

*

Take the square root of both sides.

x – 3 = ±2

Solve.

*

Example 11

Solve for y: y 2+ 2 y – 4 = 0.

Arrange in the khung of

*

Because a = 1, add

*
, or 1, lớn both sides khổng lồ complete the square.

*

Take the square root of both sides.

*

Solve.

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*

Example 12

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

*
or
*
khổng lồ both sides.

*

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.


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