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Solving quadratics by factoring

Learn how to solve quadratic equations like (x-1)(x+3)=0 and how to use factorization to solve other forms of equations.

What you should be familiar with before taking this lesson

What you will learn in this lesson

So far you have solved linear equations, which include constant terms—plain numbers—and terms with the variable raised to the first power, x1=x.
You may have also solved some quadratic equations, which include the variable raised to the second power, by taking the square root from both sides.
In this lesson, you will learn a new way to solve quadratic equations. Specifically you will learn
  • how to solve factored equations like (x1)(x+3)=0 and
  • how to use factorization methods in order to bring other equations (like x23x10=0) to a factored form and solve them.

Solving factored quadratic equations

Suppose we are asked to solve the quadratic equation (x1)(x+3)=0.
This is a product of two expressions that is equal to zero. Note that any x value that makes either (x1) or (x+3) zero, will make their product zero.
(x1)(x+3)=0x1=0x+3=0x=1x=3
Substituting either x=1 or x=3 into the equation will result in the true statement 0=0, so they are both solutions to the equation.
Now solve a few similar equations on your own.
Solve (x+5)(x+7)=0.
صرف 1 جواب چنو

Solve (2x1)(4x3)=0.
صرف 1 جواب چنو

Reflection question

Can the same solution method be applied to the equation (x1)(x+3)=6?
صرف 1 جواب چنو

A note about the zero-product property

How do we know there are no more solutions other than the two we find using our method?
The answer is provided by a simple but very useful property, called the zero-product property:
If the product of two quantities is equal to zero, then at least one of the quantities must be equal to zero.
Substituting any x value except for our solutions results in a product of two non-zero numbers, which means the product is certainly not zero. Therefore, we know that our solutions are the only ones possible.

Solving by factoring

Suppose we want to solve the equation x23x10=0, then all we have to do is factor x23x10 and solve like before!
x23x10 can be factored as (x+2)(x5).
The complete solution of the equation would go as follows:
x23x10=0(x+2)(x5)=0Factor.
x+2=0x5=0x=2x=5
Now it's your turn to solve a few equations on your own. Keep in mind that different equations call for different factorization methods.

Solve x2+5x=0.

Step 1. Factor x2+5x as the product of two linear expressions.

Step 2. Solve the equation.
صرف 1 جواب چنو

Solve x211x+28=0.

Step 1. Factor x211x+28 as the product of two linear expressions.

Step 2. Solve the equation.
صرف 1 جواب چنو

Solve 4x2+4x+1=0.

Step 1. Factor 4x2+4x+1 as the product of two linear expressions.

Step 2. Solve the equation.
صرف 1 جواب چنو

Solve 3x2+11x4=0.

Step 1. Factor 3x2+11x4 as the product of two linear expressions.

Step 2. Solve the equation.
صرف 1 جواب چنو

Arranging the equation before factoring

One of the sides must be zero.

This is how the solution of the equation x2+2x=40x goes:
x2+2x=40xx2+2x40+x=0Subtract 40 and add x.x2+3x40=0Combine like terms.(x+8)(x5)=0Factor.
x+8=0x5=0x=8x=5
Before we factored, we manipulated the equation so all the terms were on the same side and the other side was zero. Only then were we able to factor and use our solution method.

Removing common factors

This is how the solution of the equation 2x212x+18=0 goes:
2x212x+18=0x26x+9=0Divide by 2.(x3)2=0Factor.x3=0x=3
All terms originally had a common factor of 2, so we divided all sides by 2—the zero side remained zero—which made the factorization easier.
Now solve a few similar equations on your own.
Find the solutions of the equation.
2x23x20=x2+34
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے

Find the solutions of the equation.
3x2+33x+30=0
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے

Find the solutions of the equation.
3x29x20=x2+5x+16
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے