To solve the quadratic equation \(4x^2 + 10x - 3 = 0\) by completing the square, follow these steps:
**Step 1: Isolate the constant term (–3).**
Move the constant term to the right side of the equation:
\[4x^2 + 10x = 3\]
**Step 2: Divide all terms by the coefficient of \(x^2\).**
The goal is to make the coefficient of the \(x^2\) term equal to 1. Since the coefficient is 4, divide all terms by 4:
\[\frac{4x^2}{4} + \frac{10x}{4} = \frac{3}{4}\]
This simplifies to:
\[x^2 + \frac{5}{2}x = \frac{3}{4}\]
**Step 3: Add the square of half the coefficient of x to both sides.**
Find half of the coefficient of the \(x\) term (\(\frac{1}{2} \cdot \frac{5}{2} = \frac{5}{4}\)), and then square it (\(\left(\frac{5}{4}\right)^2 = \frac{25}{16}\)). This value will complete the square on the left side of the equation:
\[
\begin{align*}
x^2 + \frac{5}{2}x + \left(\frac{5}{4}\right)^2 &= \frac{3}{4} + \left(\frac{5}{4}\right)^2 \\
x^2 + \frac{5}{2}x + \frac{25}{16} &= \frac{3}{4} + \frac{25}{16}
\end{align*}
\]
**Step 4: Simplify both sides of the equation.**
Convert \(\frac{3}{4}\) to a fraction with a denominator of 16 to combine it with \(\frac{25}{16}\):
\[
\begin{align*}
x^2 + \frac{5}{2}x + \frac{25}{16} &= \frac{12}{16} + \frac{25}{16} \\
x^2 + \frac{5}{2}x + \frac{25}{16} &= \frac{37}{16}
\end{align*}
\]
**Step 5: Write the left side as a perfect square.**
The left side is now a perfect square trinomial, which can be factored into \((x + \frac{5}{4})^2\):
\[
(x + \frac{5}{4})^2 = \frac{37}{16}
\]
**Step 6: Take the square root of both sides.**
Find the square root of both sides of the equation to solve for \(x\). Remember that there will be two solutions, one positive and one negative:
\[x + \frac{5}{4} = \pm\sqrt{\frac{37}{16}}\]
**Step 7: Simplify the square root expression.**
The square root of a fraction is the square root of the numerator over the square root of the denominator:
\[x + \frac{5}{4} = \pm\frac{\sqrt{37}}{4}\]
**Step 8: Isolate \(x\).**
Subtract \(\frac{5}{4}\) from both sides to solve for \(x\):
\[x = -\frac{5}{4} \pm \frac{\sqrt{37}}{4}\]
**Step 9: Write the final solutions.**
The solutions are now in simplest form:
\[x = -\frac{5}{4} + \frac{\sqrt{37}}{4}\] or \[x = -\frac{5}{4} - \frac{\sqrt{37}}{4}\]
These can also be combined into one statement:
\[x = \frac{-5 \pm \sqrt{37}}{4}\]
So the solutions to the equation \(4x^2 + 10x - 3 = 0\) by completing the square are \(x = \frac{-5 + \sqrt{37}}{4}\) and \(x = \frac{-5 - \sqrt{37}}{4}\).
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”).
3 answers
what an illegible mess!
4x^2+10x-3=0
4(x^2 + 5/2 x) = 3
x^2 + 5/2 x = 3/4
x^2 + 5/2 x + (5/4)^2 = 3/4 + (5/4)^2
(x + 5/4)^2 = 3 + 25/16 = 37/16
x + 5/4 = ±√37/4
x = (-5±√37)/4
4x^2+10x-3=0
4(x^2 + 5/2 x) = 3
x^2 + 5/2 x = 3/4
x^2 + 5/2 x + (5/4)^2 = 3/4 + (5/4)^2
(x + 5/4)^2 = 3 + 25/16 = 37/16
x + 5/4 = ±√37/4
x = (-5±√37)/4
Apologies for the confusion. Let's go through the process of solving the equation \(4x^2 + 10x - 3 = 0\) using completing the square in a more organized fashion:
**Step 1: Divide all terms by the coefficient of \(x^2\).**
Divide every term by 4 to simplify the equation:
\[ 4x^2 + 10x - 3 = 0 \]
\[ x^2 + \frac{10}{4}x - \frac{3}{4} = 0 \]
\[ x^2 + \frac{5}{2}x - \frac{3}{4} = 0 \]
**Step 2: Add and subtract the square of half the coefficient of \(x\) on the left side.**
Calculate half of \(\frac{5}{2}\), which is \(\frac{5}{4}\), and then square it to get \(\frac{25}{16}\). Add and subtract this square inside the equation to maintain the balance:
\[ x^2 + \frac{5}{2}x + \left(\frac{5}{4}\right)^2 - \left(\frac{5}{4}\right)^2 - \frac{3}{4} = 0 \]
\[ x^2 + \frac{5}{2}x + \frac{25}{16} - \frac{25}{16} - \frac{3}{4} = 0 \]
\[ x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{25}{16} + \frac{3}{4} \]
Convert \(\frac{3}{4}\) into sixteenths to combine the fractions:
\[ x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{25}{16} + \frac{12}{16} \]
\[ x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{37}{16} \]
**Step 3: Factor the perfect square trinomial.**
The left side of the equation is a perfect square trinomial, which can be written as:
\[ \left(x + \frac{5}{4}\right)^2 = \frac{37}{16} \]
**Step 4: Solve for \(x\).**
Take the square root of both sides, keeping in mind that there is a positive and negative solution:
\[ x + \frac{5}{4} = \pm\sqrt{\frac{37}{16}} \]
\[ x + \frac{5}{4} = \pm\frac{\sqrt{37}}{4} \]
Subtract \(\frac{5}{4}\) to solve for \(x\):
\[ x = -\frac{5}{4} \pm \frac{\sqrt{37}}{4} \]
Combine the terms over a common denominator:
\[ x = \frac{-5 \pm \sqrt{37}}{4} \]
**Step 5: Write the final solutions.**
The solutions are:
\[ x = \frac{-5 + \sqrt{37}}{4} \quad \text{or} \quad x = \frac{-5 - \sqrt{37}}{4} \]
Therefore, the solutions to the equation \(4x^2 + 10x - 3 = 0\) by completing the square are \(x = \frac{-5 + \sqrt{37}}{4}\) and \(x = \frac{-5 - \sqrt{37}}{4}\).
**Step 1: Divide all terms by the coefficient of \(x^2\).**
Divide every term by 4 to simplify the equation:
\[ 4x^2 + 10x - 3 = 0 \]
\[ x^2 + \frac{10}{4}x - \frac{3}{4} = 0 \]
\[ x^2 + \frac{5}{2}x - \frac{3}{4} = 0 \]
**Step 2: Add and subtract the square of half the coefficient of \(x\) on the left side.**
Calculate half of \(\frac{5}{2}\), which is \(\frac{5}{4}\), and then square it to get \(\frac{25}{16}\). Add and subtract this square inside the equation to maintain the balance:
\[ x^2 + \frac{5}{2}x + \left(\frac{5}{4}\right)^2 - \left(\frac{5}{4}\right)^2 - \frac{3}{4} = 0 \]
\[ x^2 + \frac{5}{2}x + \frac{25}{16} - \frac{25}{16} - \frac{3}{4} = 0 \]
\[ x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{25}{16} + \frac{3}{4} \]
Convert \(\frac{3}{4}\) into sixteenths to combine the fractions:
\[ x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{25}{16} + \frac{12}{16} \]
\[ x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{37}{16} \]
**Step 3: Factor the perfect square trinomial.**
The left side of the equation is a perfect square trinomial, which can be written as:
\[ \left(x + \frac{5}{4}\right)^2 = \frac{37}{16} \]
**Step 4: Solve for \(x\).**
Take the square root of both sides, keeping in mind that there is a positive and negative solution:
\[ x + \frac{5}{4} = \pm\sqrt{\frac{37}{16}} \]
\[ x + \frac{5}{4} = \pm\frac{\sqrt{37}}{4} \]
Subtract \(\frac{5}{4}\) to solve for \(x\):
\[ x = -\frac{5}{4} \pm \frac{\sqrt{37}}{4} \]
Combine the terms over a common denominator:
\[ x = \frac{-5 \pm \sqrt{37}}{4} \]
**Step 5: Write the final solutions.**
The solutions are:
\[ x = \frac{-5 + \sqrt{37}}{4} \quad \text{or} \quad x = \frac{-5 - \sqrt{37}}{4} \]
Therefore, the solutions to the equation \(4x^2 + 10x - 3 = 0\) by completing the square are \(x = \frac{-5 + \sqrt{37}}{4}\) and \(x = \frac{-5 - \sqrt{37}}{4}\).
1 answer