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First I will describe the simpler case of the quadratic equation ax2+bx+c=0 where x=1.
To solve the quadratic equation by completing the square when a=1, first we subtract c from both sides to get x2 + bx = -c. We then rearrange the LHS of the equation into the form (x + b/2)2 however, this isn't quite right. If we expand out this form we get x2 + bx + b2 and we have an extra term of b2 from nowhere, therefore when we go into this new form we need to subtract b2 to 'complete the square'.
Now we have (x+b/2)2-b2=-c and it is just a simple case of rearranging and taking a square root to get the desired form of x= -b/2 ± sqrt(b2-c)
Now for the slightly more complicated case of where a is not 1.
With a not being 1 we cannot go from ax2 + bx to (sqrt(a)x + b/2) as when we expand it instead of getting bx we get sqrt(a)bx, so we have to use a different method. What we do is take a out of the expression as a factor to get a(x2 + bx/a + c/a) now we can do what we did before to get (x + b/2a)2 and this time we need to subtract b2/4a2. This gives us a((x + b/2a)2 + c/a - b2/4a2) multiplying a back in gives us a(x + b/2a)2 + c - b2/4a. Rearranging this for x ends up giving us the formula for solving quadratic equations, x = (-b ± sqrt(b2 - 4ac)) / 2a
I'll give an example of this latter case since all the letters can be a bit confusing when learning this technique.
example:
2x2 + 16x + 6 = 0
2(x2 + 8x + 3) = 0 taking out a factor 2
2((x + 4)2 - 13) = 0 completing the square
2(x + 4)2 - 26 = 0 factoring 2 back in
2(x + 4)2 = 26 rearranging
(x + 4)2 = 13 dividing by 2
x + 4 = ±sqrt(13) taking square root
x = -4 ±sqrt(13) rearranging for x
Verifying with the formula:
x = (-16 ±sqrt(162-48)) / 4
x = (-16 ±sqrt(208)) / 4
x = -4 ±(sqrt(208)/sqrt(16))
x = -4 ±sqrt(13)
The above shows why it is important to know the completing the square method for cases when you don't have a calculator handy, it is much easier to use the completing the square method than to work out what 162-48 is and many will not recognise that 208 is divisible by 16.
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Can all quadratics be solved by completing the square?
Completing the square is one method for solving a quadratic equation. A quadratic equation can also be solved by factoring, using the square roots or quadratic formula. Solving quadratic equations by completing the square will always work when solving quadratic equations-You can also use division or even simply take a GCF, set the quantities( ) equal to zero, and subtract or add to solve for the variable
Why do you use square roots to solve quadratic equations?
Because it's part of the quadratic equation formula in finding the roots of a quadratic equation.
X2 plus 11x plus 11 equals 7x plus 9?
X2+11x+11 = 7x+9 X2+11x-7x+11-9 = 0 x2+4x+2 = 0 Solve as a quadratic equation by using the quadratic equation formula or by completing the square: x = -2 + or - the square root of 2
What is the first step in solving this equation by completing the square?
The first step would be to find the equation that you are trying to solve!
What is the quadratic formula for?
The quadratic formula is used to solve the quadratic equation. Many equations in which the variable is squared can be written as a quadratic equation, and then solved with the quadratic formula.
Can all quadratics be solved by completing the square?
Completing the square is one method for solving a quadratic equation. A quadratic equation can also be solved by factoring, using the square roots or quadratic formula. Solving quadratic equations by completing the square will always work when solving quadratic equations-You can also use division or even simply take a GCF, set the quantities( ) equal to zero, and subtract or add to solve for the variable
Can you solve a quadratic equation by completing the square even though it is not a perfect square?
Yes, it won't be exact, but you can round the number to get a close estimate.
Name four ways to solve a quadraic equation?
Four? Factoring Graphing Quadratic Equation Completing the Square There may be more, but there's at least four.
How do you formulate quadratic equation that can be solve by extracting square roots?
By using the quadratic equation formula
What is the quadratic formula cannot be used to solve an equation if the coefficient of the equation x square term is?
The quadratic formula cannot be used to solve an equation if the coefficient of the equation x square term is what?
Solve by completing square 2x² - 6x plus 7 equals 0?
This quadratic equation has no solutions because the discriminant is less than zero.
Why do you use square roots to solve quadratic equations?
Because it's part of the quadratic equation formula in finding the roots of a quadratic equation.
When completing the square what factor must be added to the quadratic equation 2x2 3x 3 0 to solve it?
Without an equality sign and not knowing the plus or minus values of 3x and 3 the information given can't be considered to be a quadratic equation.
How do you solve a quadratic equation that will not factor?
How you solve an equation that doesn't factor is to plug a quadratic equation's format; ax2+bx+c into the quadratic formula which is x=-b+square root to (b2-4ac)/2a.
What is completing the square used for?
Completing the square is a method used to solve a quadratic function. This is a handy method when there are two instances of the same variable in the function.
How do you solve 2x squared plus x plus 2 equals 0 by completing the square?
It cannot be solved because the discriminant of the quadratic equation is less than zero
What is the solution to 4 equals 2b plus b squared?
To find the solution to this equation, you need to rearrange the terms and solve for the variable. 4 = 2b + b^2 can be rewritten as b^2 + 2b - 4 = 0. You can then solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
