What Substitution Should Be Used to Rewrite 6(X + 5)2 + 5(X + 5) – 4 = 0 as a Quadratic Equation?

Solving Quadratic Equations

A quadratic equation is an equation that could be written as

ax ² + bx + c = 0

when a 0.

At that place 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 aught on the other side.

2. Cistron.

3. Set each cistron equal to nix.

4. Solve each of these equations.

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

Example 1

Solve x ² – 6 x = 16.

Following the steps,

x ^two – vi 10 = xvi becomes x ⁱⁱ – 6 ten – 16 = 0

Factor.

( x – eight)( x + 2) = 0

Setting each factor to naught,

So to check,

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

Case 2

Solve y ² = – 6 y – 5.

Setting all terms equal to cipher,

y ² + vi y + 5 = 0

Factor.

( y + 5)( y + 1) = 0

Setting each factor to 0,

To cheque, y ² = –half dozen y – five

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

Example iii

Solve x ⁱⁱ – 16 = 0.

Factor.

To check, x ² – 16 = 0

Instance four

Solve x ⁱⁱ + 6 x = 0.

Cistron.

To check, x ² + half-dozen ten = 0

Example five

Solve 2 10 ² + 2 x – 1 = 10 ⁱⁱ + 6 x – 5.

Offset, simplify by putting all terms on one side and combining like terms.

Now, gene.

To check, two x ² + 2 x – 1 = 10 ² + 6 x – v

The quadratic formula

Many quadratic equations cannot be solved past factoring. This is generally true when the roots, or answers, are non 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 class of

ax ² + bx + c = 0

where a is the numeral that goes in front end of x ^two, b is the numeral that goes in front end of x, and c is the numeral with no variable side by side to information technology (a.m.a., "the constant").

When using the quadratic formula, yous should be aware of three possibilities. These three possibilities are distinguished by a function of the formula called the discriminant. The discriminant is the value under the radical sign, b ² – 4 air-conditioning. A quadratic equation with real numbers as coefficients tin accept the following:

1. Two different real roots if the discriminant b ² – 4 ac is a positive number.

2. One real root if the discriminant b ² – four ac is equal to 0.

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

Example 6

Solve for ten: x ² – v x = –6.

Setting all terms equal to 0,

x ² – 5 ten + 6 = 0

So substitute 1 (which is understood to be in forepart of the x ⁱⁱ), –5, and 6 for a, b, and c, respectively, in the quadratic formula and simplify.

Because the discriminant b ^two – iv air-conditioning is positive, you get two different real roots.

Example produces rational roots. In Case , the quadratic formula is used to solve an equation whose roots are non rational.

Example 7

Solve for y: y ² = –2y + 2.

Setting all terms equal to 0,

y ² + 2 y – 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.

Case 8

Solve for x: x ² + 2 x + i = 0.

Substituting in the quadratic formula,

Since the discriminant b ² – iv air conditioning is 0, the equation has one root.

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

Example 9

Solve for x: ten( 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, this equation has no solution in the existent number system.

Merely if you lot were to express the solution using imaginary numbers, the solutions would be .

Completing the square

A 3rd method of solving quadratic equations that works with both real and imaginary roots is chosen completing the square.

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

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

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

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

5. Solve the resulting equation.

Example ten

Solve for ten: ten ² – 6 x + five = 0.

Arrange in the class of

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

Accept the square root of both sides.

x – three = ±2

Solve.

Example 11

Solve for y: y ²+ 2 y – 4 = 0.

Arrange in the course of

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

Take the foursquare root of both sides.

Solve.

Example 12

Solve for ten: 2 x ² + iii 10 + 2 = 0.

Arrange in the class of

Because a ≠ 1, multiply through the equation by .

Add together or to both sides.

Accept the square root of both sides.

At that place is no solution in the existent number system. It may interest you lot to know that the completing the foursquare procedure for solving quadratic equations was used on the equation ax ⁱⁱ + 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