what is used in algebra to solve quadratic equations

Solving Quadratic Equations

A quadratic equation is an equation that could be written as

ax ⁱⁱ + bx + c = 0

when a 0.

There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.

Factoring

To solve a quadratic equation by factoring,

1. Put all terms on one side of the equal sign, leaving zero on the other side.

2. Factor.

3. Set each cistron equal to zero.

4. Solve each of these equations.

5. Cheque by inserting your answer in the original equation.

Example 1

Solve x ^two – vi x = 16.

Following the steps,

x ⁱⁱ – half-dozen x = 16 becomes x ⁱⁱ – 6 x – 16 = 0

Gene.

( x – 8)( x + ii) = 0

Setting each factor to zero,

And so to check,

Both values, 8 and –2, are solutions to the original equation.

Example ii

Solve y ² = – 6 y – 5.

Setting all terms equal to zero,

y ⁱⁱ + 6 y + 5 = 0

Factor.

( y + v)( y + 1) = 0

Setting each factor to 0,

To cheque, y ² = –6 y – v

A quadratic with a term missing is called an incomplete quadratic (as long equally the ax ² term isn't missing).

Example 3

Solve x ² – 16 = 0.

Factor.

To check, ten ⁱⁱ – 16 = 0

Example iv

Solve x ² + 6 x = 0.

Factor.

To check, x ² + six x = 0

Example 5

Solve 2 x ^two + two x – 1 = x ² + half-dozen x – five.

First, simplify past putting all terms on one side and combining like terms.

Now, factor.

To bank check, 2 x ² + 2 x – 1 = x ² + 6 x – v

The quadratic formula

Many quadratic equations cannot be solved past factoring. This is by and large true when the roots, or answers, are not rational numbers. A 2d method of solving quadratic equations involves the apply of the following formula:

a, b, and c are taken from the quadratic equation written in its general form of

ax ^two + 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 side by side to it (a.yard.a., "the constant").

When using the quadratic formula, you should be aware of three possibilities. These 3 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 existent numbers as coefficients tin have the following:

1. Two unlike real roots if the discriminant b ⁱⁱ – 4 air conditioning is a positive number.

2. 1 real root if the discriminant b ² – 4 air-conditioning is equal to 0.

3. No real root if the discriminant b ⁱⁱ – 4 ac is a negative number.

Example 6

Solve for x: 10 ⁱⁱ – 5 x = –6.

Setting all terms equal to 0,

ten ² – 5 10 + 6 = 0

Then substitute i (which is understood to be in forepart of the ten ⁱⁱ), –five, and 6 for a, b, and c, respectively, in the quadratic formula and simplify.

Because the discriminant b ² – 4 air conditioning 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 vii

Solve for y: y ² = –2y + ii.

Setting all terms equal to 0,

y ² + 2 y – 2 = 0

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

Note that the ii roots are irrational.

Example eight

Solve for ten: 10 ^two + 2 ten + ane = 0.

Substituting in the quadratic formula,

Since the discriminant b ² – 4 air-conditioning is 0, the equation has 1 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 existent number system.

Example nine

Solve for x: x( x + 2) + 2 = 0, or x ² + 2 x + two = 0.

Substituting in the quadratic formula,

Since the discriminant b ² – 4 ac is negative, this equation has no solution in the real number system.

But if y'all were to express the solution using imaginary numbers, the solutions would be .

Completing the foursquare

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

1. Put the equation into the form ax ² + bx = – c.

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

3. Using the value of b from this new equation, add to both sides of the equation to form a perfect foursquare on the left side of the equation.

4. Find the square root of both sides of the equation.

5. Solve the resulting equation.

Instance x

Solve for x: 10 ² – vi 10 + v = 0.

Arrange in the course of

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

Accept the square root of both sides.

10 – three = ±2

Solve.

Case 11

Solve for y: y ²+ 2 y – four = 0.

Arrange in the form of

Because a = 1, add , or ane, to both sides to complete the foursquare.

Take the foursquare root of both sides.

Solve.

Instance 12

Solve for x: 2 10 ^two + 3 x + 2 = 0.

Accommodate in the form of

Considering a ≠ 1, multiply through the equation past .

Add or to both sides.

Take the square root of both sides.

There is no solution in the real number system. Information technology may involvement you to know that the completing the square process for solving quadratic equations was used on the equation ax ² + bx + c = 0 to derive the quadratic formula.

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Source: https://www.cliffsnotes.com/study-guides/algebra/algebra-i/quadratic-equations/solving-quadratic-equations#:~:text=There%20are%20three%20basic%20methods,formula%2C%20and%20completing%20the%20square.&text=To%20solve%20a%20quadratic%20equation,Factor.

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