Question

Solve by quadratic formula. Give your answers in decimal form to three significant digits. Check some by calculator. $$x^{2}+2 x-7=0$$

Short answer

The solutions of the equation \(x^{2}+2 x-7=0\) are \(x \approx 1.828\) and \(x \approx -3.828\), to three significant digits.

Step-by-step solution

- Write down the quadratic formula

The quadratic formula is used to find the solutions for quadratic equations of the form \(ax^2 + bx + c = 0\). The solutions can be found using the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\).

- Identify 'a', 'b', and 'c' from the given equation

From the given quadratic equation \(x^2 + 2x - 7 = 0\), we identify the coefficients as: \(a = 1\), \(b = 2\), and \(c = -7\).

- Plug 'a', 'b', and 'c' into the quadratic formula

Substitute \(a = 1\), \(b = 2\), and \(c = -7\) into the quadratic formula to get the solutions for \(x\). \(x = \frac{{-2 \pm \sqrt{{2^2 - 4(1)(-7)}}}}{{2(1)}} = \frac{{-2 \pm \sqrt{{4 + 28}}}}{{2}} = \frac{{-2 \pm \sqrt{{32}}}}{{2}}\).

- Simplify the square root and calculate the solutions

Simplify the expression under the square root and calculate the solutions: \(\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\). Therefore, the solutions become \(x = \frac{{-2 \pm 4\sqrt{2}}}{{2}}\).

- Simplify the expression and find the decimal solutions

Divide the terms by the coefficient of \(x\), which is 2, to get the final solutions: \(x = \frac{{-2}}{{2}} \pm \frac{{4\sqrt{2}}}{{2}} = -1 \pm 2\sqrt{2}\). Calculating the decimal form of the solutions to three significant figures gives \(x \approx -1 + 2.828 = 1.828\) and \(x \approx -1 - 2.828 = -3.828\).

- Check the solutions with a calculator

Use a calculator to verify these solutions by plugging them back into the original equation and checking if they satisfy the equation. The approximations of \(1.828\) and \(-3.828\) should make the equation true.

Key concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is fundamental in algebra. A quadratic equation typically takes the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known as the coefficients, and \(x\) represents the variable we want to solve for. One of the most reliable methods for solving these equations is using the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\).

To apply this formula, simply follow these steps:

• Determine the coefficients \(a\), \(b\), and \(c\) from the equation.
• Substitute these values into the quadratic formula.
• Simplify under the square root, known as the discriminant, to find the nature of the roots (whether they are real or complex).
• Calculate the numerical values of the roots by following the operations in the formula.

Using these steps will lead to the roots of the quadratic equation, which are the values of \(x\) that make the equation true. It's like finding the exact points where a parabola, which is the graphical representation of a quadratic equation, crosses the x-axis.

Quadratic Equation Roots

The roots of a quadratic equation are the solutions to the equation; they are the points at which the associated parabola intersects the x-axis. There can be up to two real roots, which can be equal (resulting in one unique solution) or distinct, depending on the discriminant \(\sqrt{{b^2 - 4ac}}\).

When the discriminant is positive, there will be two distinct real roots. If it is zero, there is one repeated root, known as a double root. Finally, if the discriminant is negative, the equation has no real roots, but two complex roots. For the given equation \(x^2 + 2x - 7 = 0\), we find two real roots after applying the quadratic formula. The roots are \(x = -1 \pm 2\sqrt{2}\), which simplifies to two different numbers, indicating where the parabola touches the axis.

Significant Digits

Significant digits, also known as significant figures, are a way of expressing precision in numerical answers. When solving mathematical problems, it's important to round your final answer to the appropriate number of significant digits to accurately reflect the precision of the given data.

For instance, when asked to provide answers in decimal form to three significant digits, as in the case of our quadratic roots \(-1 \pm 2\sqrt{2}\), we express the solutions as \(x \approx 1.828\) and \(x \approx -3.828\) after rounding. Using significant figures correctly is crucial as it communicates the degree of certainty in the measurements or calculations you've made, ensuring that the answers are both accurate and practical.