Question

Solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ 2 x^{2}+5\left(x^{2}-2\right)=29 $$

Short answer

Therefore, the exact solution to the quadratic equation is \(x = ± \sqrt{39 / 7}\), and the rounded decimal solution is \(x = ± 2.51\)

Step-by-step solution

Simplify the quadratic equation

Begin by simplifying the equation and combining like terms: \[ 2x^{2} + 5x^{2} - 10 = 29 \] After simplifying, we get: \[ 7x^{2} - 10 = 29 \]

Solve for the square of \(x\)

To isolate \(x^{2}\), we must get rid of -10 from the left hand side by adding 10 to both sides of the equation, then divide each side by 7: \[ x^{2} = (29 + 10) / 7 \] This simplifies to: \[ x^{2} = 39 / 7 \]

Find the square root of both sides in order to isolate \(x\)

The next step is to find the square root of both sides of the equation: \[ x = ± \sqrt{39 / 7} \]

Provide the decimal approximation to two decimal places

Finally, the decimal approximation (rounded to two decimal places) is: \[ x = ± 2.51 \]

Key concepts

Extracting Square Roots

The method of extracting square roots is a straightforward technique to solve a quadratic equation that has the general form of ax^2 = c. It involves isolating the term containing the square of the variable and then applying the square root to both sides of the equation.

Let's illustrate this with a step involved in our given exercise. After simplifying the equation, we arrived at x^2 = 39/7. The next step required us to extract the square roots to solve for x. The mathematical operation applied here is taking the square root of both sides of the equation, resulting in x = ±√(39/7). Bear in mind that the square root has two values: one positive and one negative, hence the symbol '±' preceding the square root.

By grasping the concept of extracting square roots, students can solve a variety of quadratic equations swiftly and effectively when the equation is properly structured. This approach, while not always suitable for every quadratic equation, is a valuable tool within a student's mathematical arsenal.

Simplifying Quadratic Equations

Simplifying quadratic equations is a crucial step before any method of solution can be applied. It involves combining like terms and eliminating any unnecessary complexity within the equation. Through simplification, we can bring the equation to its standard form ax^2 + bx + c = 0, making it easier to solve.

In our exercise, we began by expanding and then combining like terms: the 2x^2 and 5x^2 are like terms and were combined to 7x^2. We also had to move the constant term to the other side of the equation by adding 10 to both sides, resulting in 7x^2 = 39. Simplifying a quadratic equation requires careful attention to algebraic principles such as distribution, combining like terms, and maintaining the balance of the equation through legal operations on both sides.

Understanding and mastering the process of simplifying quadratic equations empower students to tackle a wide range of quadratic problems and is often the first and most important step towards finding the solution.

Decimal Approximation

Decimal approximation is the process of expressing a number in a more practical, rounded form while maintaining a compromise between accuracy and simplicity. In the context of solving quadratic equations, after finding the exact solutions, it can be useful to convert these to decimal form, particularly for real-world applications or comparison purposes.

In the final step of our solution, we focused on obtaining the decimal approximation of ±√(39/7). Using a calculator, we found that the square root of 39/7 is approximately 2.51 when rounded to two decimal places. The '±' symbol indicates that we should consider both the positive and negative forms of our solution, thus x ≈ ±2.51. Decimal approximation is especially useful when the exact answer involves irrational numbers, which can't be expressed as a simple fraction or involve long decimal expansions. In such cases, rounding to a fixed number of decimal places makes the solution more understandable and usable.