- Isolate the constant and factor the quadratic expression to get \(5(x^2 + 4x) = 7\).
- Add the square value to both sides, noting the multiplier, to get \(5(x^2 + 4x + 4) = 7 + 20\).
- Simplify to identify the square root step, resulting in \(x + 2 = \pm \sqrt{\frac{27}{5}}\).
- The three correct options are \(5(x^2 + 4x) = 7\), \(5(x^2 + 4x + 4) = 7 + 20\), and \(x + 2 = \pm \sqrt{\frac{27}{5}}\).
Explanation
- Analysis of the Problem
We are given the quadratic equation \(5x^2 + 20x - 7 = 0\). To solve this by completing the square, we need to transform it into the form \((x + h)^2 = k\). We will follow the standard procedure:
- Isolate the constant term on the right side.
- Factor out the leading coefficient from the x-terms.
- Determine the constant term required to complete the square inside the parenthesis.
- Add the balanced value to the right side of the equation.
- Factor the perfect square trinomial and solve.
- Isolating and Factoring
We begin with the original equation:
\[5x^2 + 20x - 7 = 0.\]
First, add 7 to both sides to isolate the variable terms:
\[5x^2 + 20x = 7.\]
Next, factor out the coefficient of \(x^2\), which is 5, from the terms on the left:
\[5(x^2 + 4x) = 7.\]
This step matches one of the provided options: \(5(x^2 + 4x) = 7\).
- Completing the Square
To complete the square for \(x^2 + 4x\), we take half of the coefficient of \(x\) and square it: \(\left(\frac{4}{2}\right)^2 = 2^2 = 4\).
We add 4 inside the parenthesis. Since there is a factor of 5 outside the parenthesis, we must add \(5 \times 4 = 20\) to the right side of the equation to keep it balanced:
\[5(x^2 + 4x + 4) = 7 + 20.\]
This matches the option: \(5(x^2 + 4x + 4) = 7 + 20\).
- Solving the Equation
Now we rewrite the left side as a perfect square and simplify the right side:
\[5(x + 2)^2 = 27.\]
Divide both sides by 5:
\[(x + 2)^2 = \frac{27}{5}.\]
Taking the square root of both sides gives:
\[x + 2 = \pm \sqrt{\frac{27}{5}}.\]
This matches the option: \(x + 2 = \pm \sqrt{\frac{27}{5}}\). We have identified the three correct steps.
Examples
Completing the square is a powerful method used in physics and engineering to analyze projectile motion. For example, if you throw a ball, its height over time is often modeled by a quadratic equation. By completing the square, you can easily identify the vertex of the parabola, which represents the maximum height the ball reaches and the time at which it happens. It is a fundamental tool for understanding parabolic trajectories in real-world dynamics.