Solving the quadratic equation 4x^2 + 10x - 3 = 0 using the method of completing the square involves several steps. Here's a step-by-step guide explaining each stage of the process.
Step 1: Divide by the leading coefficient to make the x^2 coefficient equal to 1.
The leading coefficient is the number in front of the x^2 term, which in this case is 4. We want to divide the whole equation by 4 to simplify the quadratic equation.
4x^2 + 10x - 3 = 0
⇒ x^2 + (10/4)x - (3/4) = 0
⇒ x^2 + (5/2)x - 3/4 = 0
Step 2: Move the constant term to the right side of the equation.
We want to isolate the terms containing x on one side. To do this, let's add 3/4 to both sides of the equation to move the constant term.
x^2 + (5/2)x - 3/4 = 0
⇒ x^2 + (5/2)x = 3/4
Step 3: Complete the square on the left side.
To complete the square, find a number that, when added and subtracted to the left side, will convert the left side into a perfect square trinomial. This number is the square of half the coefficient of the x-term.
Half of 5/2 is 5/4, and the square of 5/4 is (5/4)^2.
Now, add and subtract (5/4)^2 to the left side to create a perfect square trinomial.
x^2 + (5/2)x + (5/4)^2 - (5/4)^2 = 3/4
⇒ x^2 + (5/2)x + 25/16 - 25/16 = 3/4
Step 4: Simplify and add (5/4)^2 to both sides of the equation.
We group the trinomial and combine the constants.
(x^2 + (5/2)x + 25/16) - 25/16 = 3/4 + 25/16
First, calculate 3/4 + 25/16. It will be helpful to use a common denominator:
(3/4) * (4/4) = 12/16
So, 3/4 + 25/16 = (12/16) + (25/16) = 37/16
The equation now is:
(x^2 + (5/2)x + 25/16) = 37/16
Step 5: Write the left side as a square and solve for x.
The left side of the equation is a perfect square trinomial, and it can be expressed as a square of a binomial:
(x + 5/4)^2 = 37/16
Step 6: Take the square root on both sides of the equation to solve for x.
√((x + 5/4)^2) = ±√(37/16)
When you take the square root of a squared expression, you get the absolute value, so you must consider both the positive and negative square roots.
x + 5/4 = ±√(37/16)
Let's calculate √(37/16) which equals ±√37/4.
x + 5/4 = ±√37/4
Step 7: Isolate x.
Now, we solve for x by subtracting 5/4 from both sides.
x = -5/4 ± √37/4
x = (-5 ± √37)/4
Here you have the solutions to the original quadratic equation using the method of completing the square. There are two solutions because of the ± sign indicating the positive and negative roots. The solutions are:
x = (-5 + √37)/4
x = (-5 - √37)/4
And that's how you complete the square to solve the equation 4x^2 + 10x - 3 = 0.
Write a step-by-step guide on how to solve the following problem using completing the square: 4x^2+10x-3=0. The guide should show both the algebra and written explanations for what is occurring at each step, and the guide should use the proper mathematical notation
1 answer