Problem 2 Solve each equation using the Qu... [FREE SOLUTION] (2024)

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Solving quadratic equations is a foundational skill in algebra. A quadratic equation is generally written in the form of ax^2 + bx + c = 0, where a, b, and c are constants. The most common methods to solve these equations are factoring, completing the square, using the quadratic formula, or graphing. The quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), is especially handy when the equation is not easily factorable. By substituting the coefficients of the quadratic equation into this formula, we can efficiently find the roots or solutions of the equation.

For example, to solve the equation \( x^2 + 8x + 12 = 0 \), we start by identifying the coefficients a=1, b=8, and c=12. We can then apply these values to the quadratic formula to obtain the equation's roots. This formula offers a systematic approach that works for any quadratic equation, regardless of whether it can be easily factored, making it an invaluable tool in a student’s mathematical toolkit.

The roots of a quadratic, also known as its solutions or x-intercepts, are the values of x that make the quadratic equation equal zero. These solutions can be real or complex numbers and there can be one or two distinct roots, depending on the discriminant (the part under the square root in the quadratic formula, \( b^2 - 4ac \) ).

The discriminant tells us the nature of the roots:

• If it's positive, there are two real and distinct roots.
• If it's zero, there is one real repeated root.
• If it's negative, there are two complex roots.

Using the provided equation \( x^2 + 8x + 12 = 0 \), after applying the quadratic formula, we find that the discriminant is \( 8^2 - 4\cdot1\cdot12 = 16 \), a positive number. This means our equation has two real and distinct roots, which are the solutions to the equation. Simplified further, these roots are x = -2 and x = -6.

In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the coefficients are the numerical factors that multiply the respective terms of the equation. The coefficient a is in front of the \( x^2 \) term, b is the coefficient of the \( x \) term, and c is the constant term. These coefficients are crucial as they determine the shape and position of the parabola when graphed, and most importantly, they are directly used when applying the quadratic formula to solve for the roots.

In the exercise \( x^2 + 8x + 12 = 0 \), a is 1 (since there’s no number, it's understood to be 1), b is 8, and c is 12. When using the quadratic formula, these values are plugged in precisely as they appear in the equation to determine the equation's roots. Understanding the role of these coefficients is essential for correctly applying various methods to analyze and solve quadratics.

Simplifying square roots is a necessary step in solving quadratic equations using the quadratic formula. When confronted with an expression under a square root, the goal is to find the prime factorization and pair the factors to simplify the root as much as possible. In cases where the number under the square root is a perfect square, its square root will be a rational number.

Consider the square root we encountered in our equation, \( \sqrt{16} \). Since 16 is a perfect square, the square root of 16 is simply 4. If the number under the square root was not a perfect square, the process would involve finding the largest square factor of the number and simplifying from there. The ability to simplify square roots efficiently can greatly expedite the process of solving quadratic equations, especially when using the quadratic formula.