Question

Find a possible pair of integer values for \(a\) and \(c\) so that the quadratic equation has the given solution(s). Then write the equation. \(a x^2+6 x+c=0\); two real solutions

Short answer

One possible pair of integer values for 'a' and 'c' could be 1 and 5, respectively, and the quadratic equation would then be \(x^2+6x+5=0\). Remember that other pairs of 'a' and 'c' that satisfy the condition \(ac<9\) will also be possible solutions.

Step-by-step solution

Determine the discriminant formula

The formula for the discriminant in a quadratic equation is \(b^2 - 4ac\).

Insert the given value of b

We have the coefficient \(b = 6\), so the formula changes to \(36 - 4ac > 0\).

Solve for ac

Now, from the equation \(36 - 4ac > 0\), we get \(ac < 9\). This means the product of 'a' and 'c' should be less than 9.

Choose the possible integer pair

The integer pairs that satisfy \(ac < 9\) could be many. An example could be \(a = 1, c = 5\), but any number pair that multiplies to less than 9 where both are integers would work.

Write the quadratic equation

Knowing the values for 'a' and 'c', we can write the quadratic equation. In our case, a possible quadratic equation could be \(x^2+6x+5=0\).

Key concepts

Discriminant of a Quadratic Equation

Understanding the discriminant of a quadratic equation is crucial in determining the nature of its solutions. The discriminant is part of a larger formula used to solve quadratic equations, and it provides insight into whether a quadratic equation will have real or complex solutions, and if the solutions are real, whether they are distinct or the same.

A quadratic equation in the standard form is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The discriminant \( D \) for such an equation is given by the formula \( D = b^2 - 4ac \).

The value of the discriminant tells us the following:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution (also called a repeated or double root).
• If \( D < 0 \), the equation has no real solutions; instead, it has two complex solutions.

For the exercise given, we are looking for integer values such that the discriminant is positive, meaning that the equation will have two distinct real solutions. Since \( b = 6 \) in \( ax^2 + 6x + c = 0 \), substituting into the discriminant formula gives us \( 36 - 4ac > 0 \) to ensure two real solutions, leading to the condition \( ac < 9 \).

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. There are several methods to find the roots or solutions of a quadratic equation, such as factoring, completing the square, using the quadratic formula, and graphing. The most suitable method depends on the form of the equation and the coefficients involved.

When the quadratic equation can be easily factored, this method is often the quickest. For more complex cases, the quadratic formula is a powerful tool that provides solutions regardless of the form of the quadratic equation. To complete the square, one must manipulate the equation to form a perfect square trinomial, which can then be solved by taking the square root of both sides.

In our textbook exercise example, factoring might be the simplest method once integer values for \( a \) and \( c \) that satisfy the discriminant condition \( ac < 9 \) are found. If factoring is difficult or impossible, applying the quadratic formula with the determined values of \( a \) and \( c \) always works.

Quadratic Formula

The quadratic formula is derived from the process of completing the square and gives an exact solution for the roots of a quadratic equation. It is a versatile and widely used method for finding the solutions of any quadratic equation. The formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

The symbol \( \pm \) indicates that there are two solutions: one when you add the square root of the discriminant (root plus) and the other when you subtract it (root minus). These solutions correspond to the x-intercepts of the quadratic function's graph or the points at which the parabola crosses the x-axis.

Applying the Quadratic Formula

Using the quadratic formula involves substituting the values of \( a \) , \( b \) , and \( c \) from the equation into the formula and simplifying. It is important to pay attention to the signs of the coefficients and to perform each step carefully to ensure an accurate solution.

In relation to the exercise provided, after selecting appropriate integer values for \( a \) and \( c \) such that the discriminant is positive, one could use the quadratic formula to determine the specific solutions for the equation \( ax^2 + 6x + c = 0 \) or check the correctness of the integer pairs chosen.