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
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.