Solving Quadratic Equations

A quadratic equation is an equation that could be written as

ax² + bx + c = 0

when a 0.

There are three basic methods for solving quadratic equations:

I. Factoring
II. Using the quadratic formula, and
III. 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.

⇒ Factor.

⇒ Set each factor equal to zero.

⇒ Solve each of these equations.

⇒ Check by inserting your answer in the original equation

Example 1

Solve x² – 6 x = 16.

Following the steps,

x² – 6x = 16 becomes x² – 6 x – 16 = 0

Factor of 16 = -8 and +2 Ie (8 x 2).



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 form of


ax² + bx + c = 0

where a is the numeral that goes in front of x², b is the numeral that goes in front of x, and c is the numeral with no variable next to it.

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, b² – 4 ac. A quadratic equation with real numbers as coefficients can have the following:

⇒ Two different real roots if the discriminant b² – 4 ac is a positive number.

⇒ One real root if the discriminant b² – 4 ac is equal to 0.

⇒ No real root if the discriminant b² – 4 ac is a negative number.

Example 2

Solve for x: x² – 5x = –6.

Setting all terms equal to 0,

x² – 5x + 6 = 0

Then substitute 1 (which is understood to be in front of the x²), –5, and 6 for a, b, and c, respectively, in the quadratic formula and simplify.



Because the discriminant b² – 4 ac is positive, you get two different real roots.

Example produces rational roots. In Example , the quadratic formula is used to solve an equation whose roots are not rational.

Example 3

Solve for y,when  y² = –2y + 2.

Setting all terms equal to 0,

y² + 2y – 2 = 0

Then substitute 1, 2, and –2 for a, b, and c, respectively, in the quadratic formula and simplify.



Note that the two roots are irrational.

Example 4

Solve for x: x² + 2x + 1 = 0.

Substituting in the quadratic formula,



Since the discriminant b² – 4 ac is 0, the equation has one root.

The quadratic formula can also be used to solve quadratic equations whose roots are imaginary numbers, that is, they have no solution in the real number system.

Completing the square

A third method of solving quadratic equations that works with both real and imaginary roots is called completing the square.

Put the equation into the form ax² + bx = – c.

Make sure that a = 1 (if a ≠ 1, multiply through the equation by equation before proceeding).

Using the value of b from this new equation, add equation to both sides of the equation to form 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 5

Solve for x: x² – 6x + 5 = 0.

Arrange in the form of



Because a = 1, add equation, or 9, to both sides to complete the square.



Take the square root of both sides.

x – 3 = ±2

Solve.



Example 6

Solve for y: y² + 2y – 4 = 0.

Arrange in the form of



Because a = 1, add equation , or 1, to both sides to complete the square.



Take the square root of both sides.



Solve.




Example 7

Solve for x: 2x² + 3x + 2 = 0.

Arrange in the form of



Because a ≠ 1, multiply through the equation by







Take the square root of both sides.



There is no solution in the real number system. It may interest you to know that the completing the square process for solving quadratic equations was used on the equation ax² + bx + c = 0 to derive the quadratic formula.