How do you convert a quadratic equation from standard form into factored form?
The Correct Answer and Explanation is:
To convert a quadratic equation from standard form ( ax^2 + bx + c = 0 ) to factored form ( a(x – r_1)(x – r_2) = 0 ), follow these steps:
Steps for Conversion:
- Factor out the greatest common factor (GCF) if possible.
- Identify coefficients: ( a ), ( b ), and ( c ).
- Find two numbers that multiply to ( a \cdot c ) (the product of the leading coefficient and constant term) and add to ( b ) (the coefficient of ( x )).
- Split the middle term using these two numbers to rewrite the quadratic as four terms.
- Group terms in pairs and factor each pair.
- Factor out the common binomial factor to get the factored form.
If the quadratic cannot be factored easily, use the quadratic formula to find the roots (( r_1 ) and ( r_2 )), and write the equation as ( a(x – r_1)(x – r_2) = 0 ).
Example:
Convert ( 2x^2 + 5x + 3 = 0 ) to factored form.
- Identify coefficients: ( a = 2 ), ( b = 5 ), ( c = 3 ).
- Find two numbers that multiply to ( 2 \cdot 3 = 6 ) and add to ( 5 ). These are ( 2 ) and ( 3 ).
- Split the middle term: ( 2x^2 + 2x + 3x + 3 = 0 ).
- Group terms: ( (2x^2 + 2x) + (3x + 3) = 0 ).
- Factor each group: ( 2x(x + 1) + 3(x + 1) = 0 ).
- Factor the common binomial: ( (2x + 3)(x + 1) = 0 ).
The factored form is ( (2x + 3)(x + 1) = 0 ).
Explanation:
Factoring simplifies quadratic equations to find their roots efficiently. The roots represent the values of ( x ) that make the equation zero, which are also the points where the graph intersects the ( x )-axis. This method is vital in algebra as it connects polynomial structure to its solutions and applications in real-world problems like projectile motion and optimization.