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The Quadratic Formula
(Explanation)When ax2+bx+c=0(a≠0) then x= [ The Square Root Method ]
When x2=a, then x=±
(If a<0, the solution sets express complex imaginary numbers.) [ Completing the Square ]
By completing the square, you can use the Square Root method for "any" quadratic equation.
i.e. You can solve "any" quadratic equation by Completing the Square. When (x−a)2=b, then x−a=± [ The Quadratic Formula ]
The method of completing the square is a little complicated.
The Quadratic Formula briefly summarizes the result of completing the square. When ax2+bx+c=0(a≠0), then x= |
(Derivation of Quadratic Formula) When ax2+bx+c=0(a≠0) Divide both sides by a x2+x+=0 Factor out 2 from the x-term. x2+2x+=0 Complete the square x2+2x+()2+=()2 x2+2x+()2=()2−= (x+)2= Take the Square Root x+=±=± Therefore x=−± x= |
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ExampleSolve the following equations for x.
1.3x2+7x+1=0
Identify a, b, and c.→ a=3, b=7, c=1
Substitute the value of a,b and c into the Formula.→ x=
Simplify→ x=
2.x2+4x−6=0
Identify a, b, and c.→ a=1, b=4, c=−6
Substitute the value of a,b and c into the Formula.→ x=
Simplify→ x===−2±
3.−2x2+5x+3=0
You had better multiply both sides of the equation by −1 in order to change the coefficient of x2-term to positive value.
Otherwise you will encounter a little complicated fraction like the following.
Multiply both sides of the equation by −1In such a fraction many students tend to make mistakes with signs. → 2x2−5x−3=0
Identify a, b, and c.→ a=2, b=−5, c=−3
Substitute the value of a,b and c into the Formula.→ x=
Simplify→ x==
x=3, − |
4.(x−1)(x+3)=4
First, move all terms to the left side and rewrite the equation in the standard form.→ x2+2x−3=4 x2+2x−7=0 Identify a, b, and c. → a=1, b=2, c=−7
Substitute the value of a,b and c into the Formula.→ x=
Simplify→ x===−1±2
Solving Quadratic Equations by Formula (Summary)
(Step *)Move all terms to the left side and rewrite the equation in the standard form if needed.
e.g.x2+2x−3=4 → x2+2x−7=0
(Step *)Multiply both sides of the equation by −1 if the coefficient of x2-term is the negative value.
e.g.−2x2+3x−5=0 → (−1)×(−2x2+3x−5)=(−1)×0
→ 2x2−3x+5=0 Step 1Identify a, b, and c.
e.g.3x2−4x+5=0 → a=3,b=−4,c=5
Step 2Substitute the value of a,b and c into the Formula.
e.g.a=3,b=−4,c=5 → x=
Step 3Simplify.
e.g.x= → x==−2±
e.g.x= → x=3, −
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Quadratic Formula Calculator You can use these calculators to find the solutions to your quadratic equation.
• Exact solutions
(Often used in math textbook or in class) |
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• Approximate solutions
(Often used in science or in computer science) |
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QuestionSolve the following equations for x. (Find the exact solutions.) |
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