Question

Solve the quadratic equation. $$c^{2}+2 c-26=0$$

Short answer

The solutions to the quadratic equation are \( x = -1 + 5\sqrt{1.08} \) and \( x = -1 - 5\sqrt{1.08} \).

Step-by-step solution

Identify the coefficients

Firstly, identify the coefficients of the quadratic equation. In this case: \(a = 1\), \(b = 2\) and \(c = -26\).

Apply the quadratic formula

Substitute the coefficients into the quadratic formula \((x = (-b ± \sqrt{b^2 - 4ac}) / (2a) \). This results in: \( x = (-2 ± \sqrt{2^2 - 4*1*(-26)}) / (2*1) \).

Simplify

Simplify that formula further: \( x = (-2 ± \sqrt{4 + 104}) / 2 = (-2 ± \sqrt{108}) / 2 \). Now, \( \sqrt{108} \) can be simplified to \( 10\sqrt{1.08} \), hence, re-write as: \( x = (-2 ± 10\sqrt{1.08}) / 2 \). \nFinally, divide through by 2: \( x = -1 ± 5\sqrt{1.08} \).

Key concepts

Quadratic Formula

Understanding the quadratic formula is essential for solving quadratic equations. It is a foolproof method that applies to any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

To apply the formula, it’s important to ‘Identify the coefficients’ of a, b, and c from the quadratic equation, which are the numerical values in front of the variable terms. Once you have identified these values, substitute them into the formula. The term \(b^2 - 4ac\) under the square root is called the discriminant, and it determines the number and type of solution the equation will have.

If the discriminant is positive, there will be two real solutions. If it is zero, you can expect one real solution, and if negative, there are two complex solutions. After calculating the discriminant, take the square root, add or subtract it from -b, and finally, divide by \(2a\) to get the solutions. This method ensures that you can find the roots of any quadratic equation.

Simplifying Square Roots

Simplifying square roots can be a stumbling block for students mastering quadratic equations. The purpose is to express the square root in its simplest form. To simplify \(\sqrt{108}\), for example, find the largest perfect square factor of 108, which is 36. So, \(\sqrt{108}\) can be re-written as \(\sqrt{36 \times 3}\), or \(6\sqrt{3}\).

Be attentive to the fact that simplifying square roots involves factoring the number inside the root and retrieving the largest square factors outside. A perfect square is an integer that can be expressed as the square of another integer, like 4, 9, 16, and so on. Always break down the number inside the root into these squares when possible, to simplify the expression for further calculations. This step can massively streamline the solving process of a quadratic equation when using the quadratic formula.

Identifying Coefficients

Identifying coefficients in a quadratic equation is the first step to solving it. Coefficients are the numbers multiplying the variable. In the quadratic equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are the coefficients of the terms \(x^2\), \(x\), and the constant term, respectively.

In the provided exercise \(c^2 + 2c - 26 = 0\), the coefficient \(a\) is 1 (since \(c^2\) is the same as \(1c^2\)), \(b\) is 2, and \(c\) is -26. It is crucial to correctly identify these coefficients because they are necessary inputs for the quadratic formula. Errors in identifying the coefficients can lead to the wrong discriminant and ultimately incorrect solutions. Always check that \(a\), \(b\), and \(c\) correspond to the correct parts of the equation before proceeding with solving.