In the previous section, you used the method of completing the square for expressions of the form x2 + bx +c, where the coefficient of the x2-term was 1. But what would you do if you encountered an expression like the one shown below?
If you were to factor a 2 out of each term, what would be the resulting expression?
Click through the animation below to show the steps of solving the problem above.
How is this process similar to what you did when the coefficient of the x2-term was 1?
How is this process different from what you did when the coefficient of the x2-term was 1?
If the coefficient of the x2-term were 3 instead of 2, what would you need to do differently?
Interactive popup. Assistance may be required.
Instead of factoring out a 2, you would need to factor out a 3.
If the polynomial is of the form ax2 + bx + c, generalize how you would complete the square in order to generate a vertex form of the same expression, a(x – h)2 + k.
Interactive popup. Assistance may be required.
If the coefficient of the x2-term were 1 over 3 1 3 instead of 2, what would you need to do differently?
Interactive popup. Assistance may be required.
Instead of factoring out a 2, you would need to factor out a 1 over 3 1 3.
Interactive popup. Assistance may be required.
Factor the coefficient of the x2-term (in this case, 2) from the entire polynomial before completing the square. Be sure to redistribute the 2 to both the squared binomial and the constant term.
Interactive popup. Assistance may be required.
2x2 - 8x + 16
2[x2 - 4x + 8]
2[x2 - 4x + (2)2 + 8 - (2)2]
2[(x - 2)2 + 8 - 4]
2[(x - 2)2 + 4]
2(x - 2)2 + 8
Interactive popup. Assistance may be required.
Factor the coefficient of the x2-term (in this case, 4) from the entire polynomial before completing the square. Be sure to redistribute the 4 to both the squared binomial and the constant term.
Interactive popup. Assistance may be required.
4x2 + 20x - 12
4[x2 + 5x - 3]
4[x2 + 5x + (2.5)2 - 3 - (2.5)2]
4[(x + 2.5)2 -3 - 6.25]
4[(x + 2.5)2 - 9.25]
4(x + 2.5)2 - 37
Use the method of completing the square to rewrite 3x2 + 18x = 12 in vertex form.
Interactive popup. Assistance may be required.
Factor the coefficient of the x2-term (in this case, 3) from the entire left member of the equation before completing the square. Be sure to redistribute the 3 to the squared binomial and make sure that you have added the same quantity to both sides of the equation in order to preserve the equality.
Interactive popup. Assistance may be required.
3x2 + 18x = 12
3[x2 + 6x] = 12
3[x2 + 6x + (3)2] = 12 + 3(3)2
3(x + 3)2 = 12 + 27
3(x + 3)2 = 39