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 (do NOT use “^”, “/”, “sqrt”).
To solve the quadratic equation 4x² + 10x - 3 = 0 by completing the square, follow these steps:
**Step 1: Divide the entire equation by the coefficient of x² to get the leading coefficient to be 1.**
Divide every term by 4:
(4x²/4) + (10x/4) - (3/4) = 0
This simplifies to:
x² + (5/2)x - 3/4 = 0
**Step 2: Move the constant term to the right side of the equation.**
Add 3/4 to both sides of the equation to isolate the x terms:
x² + (5/2)x = 3/4
**Step 3: Complete the square for the x term on the left side of the equation.**
To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation. The coefficient in front of x is 5/2, so half of that is 5/4. Squaring 5/4:
(5/4)² = 25/16
Now, add 25/16 to both sides of the equation:
x² + (5/2)x + 25/16 = 3/4 + 25/16
You need to have a common denominator to combine the terms on the right side of the equation. Since 4 is the denominator of 3/4, multiply numerator and denominator of it by 4 to match the denominator of 25/16:
3/4 + 25/16 = (3*4)/(4*4) + 25/16
= 12/16 + 25/16
= (12 + 25)/16
= 37/16
So the equation is now:
x² + (5/2)x + 25/16 = 37/16
**Step 4: Factor the perfect square trinomial on the left side.**
The left side of the equation is now a perfect square trinomial, which can be factored into:
(x + 5/4)² = 37/16
**Step 5: Solve for x by taking the square root of both sides.**
To solve for x, you need to get rid of the squared term by taking the square root of both sides. Remember to consider both the positive and negative square roots:
√(x + 5/4)² = ±√(37/16)
x + 5/4 = ±√(37/16)
Now simplify the square root on the right side:
x + 5/4 = ±√37/4
**Step 6: Isolate x by subtracting 5/4 from both sides of the equal sign.**
To solve for x, subtract 5/4 from both sides:
x = -5/4 ± √37/4
**Step 7: Simplify the expression if possible.**
You can leave it like that or if you want to separate the fraction:
x = -5/4 ± (√37)/4
**Step 8: Write the final solutions.**
The solution to the quadratic equation by completing the square method is:
x = (-5 ± √37)/4
These are the two possible values for x that satisfy the original equation.