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. \(a x^2-8 x+c=0\); two real solutions

Short answer

Possible integer pairs for a and c that satisfy the inequality \(ac < 16\) include (1, 1), or (2, 2), or (2, 3) or (3, 1) etc. Once we have this pair we substitute it in the quadratic equation for example: If a=1, c=1 we get \( x^2 - 8x + 1 = 0\) ; if a=2, c=3 we get \(2x^2 - 8x + 3 = 0\).

Step-by-step solution

Determining the Discriminant

The discriminant, D, of a quadratic equation is given by \(D = b^2 - 4ac\). For the given quadratic equation \(ax^2 - 8x + c = 0\), b = -8. We need squares b and get 64.

Setting Up an Inequality with the Discriminant

To find a pair of integer values for a and c such that D > 0(for two real solutions) we can set an inequality with D. Substituting in the equation: \(64 - 4ac > 0\). When simplified we get \( 4ac < 64\), or \(ac < 16\)

Finding Pairs of a and c

Possible integer pairs for a and c that satisfy the inequality \(ac < 16\) include (1, 1), or (2, 2), or (2, 3) or (3, 1) etc. Once we have this pair we substitute it in quadratic equation.

Writing out the quadratic equation

Substitute a and c in our quadratic equation. For instance, if a=1, c=1 we get \( x^2 - 8x + 1 = 0\) ; if a=2, c=3 we get \(2x^2 - 8x + 3 = 0\).

Key concepts

Discriminant

The discriminant is a significant tool in determining the nature of the solutions of a quadratic equation, which is typically of the form \(ax^2 + bx + c = 0\). The discriminant is symbolized by \(D\) and is calculated using the formula \(D = b^2 - 4ac\). This value tells us how many and what type of solutions a quadratic equation has.

• If \(D > 0\), the quadratic equation has two distinct real solutions.
• If \(D = 0\), the quadratic equation has exactly one real solution, considered a repeated or double root.
• If \(D < 0\), the equation has no real solutions but two complex solutions.

In the original exercise, we desire two real solutions for the quadratic equation \(ax^2 - 8x + c = 0\). We start by recognizing \(b = -8\), which gives us \(b^2 = 64\). Therefore, we set up the equation \(64 - 4ac > 0\) to ensure that the discriminant is positive, indicating two real solutions.

Integer Solutions

Finding integer solutions for variables within a quadratic equation requires consideration of the discriminant outcome and possible values for \(a\) and \(c\). The task involves identifying integers \((a, c)\) that satisfy the condition \(ac < 16\), derived from the need for a positive discriminant in the form \(64 - 4ac > 0\).

To break it down:

• Calculate \(ac\) values that are less than 16.
• Identify integer pairs \((a, c)\) that fit this criterion.

This means systematically testing pairs such as \((1, 1)\), \((2, 2)\), \((2, 3)\), and \((3, 1)\). Each pair must maintain the condition \(ac < 16\) to ensure two real solutions exist when substituted back into the quadratic equation formula. For example, the pair \((2, 3)\) results in \(ac = 6\), well within the threshold, making it a valid choice.

Real Solutions

Real solutions in a quadratic equation refer to the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\) and are real numbers. Ensuring two real solutions involves maintaining a positive discriminant (\(D > 0\)) so that \(b^2 - 4ac\) remains greater than zero.

The factorization or quadratic formula methods can solve the equation once integers \(a\) and \(c\) meet the condition \(ac < 16\). These methods allow us to express the solutions in terms of radicals or exact numbers, revealing where the graph of the quadratic function intersects the x-axis.

• Quadratics with two real solutions will show as two points where the graph crosses the x-axis.
• These intersections are found using the formula \(x = \frac{-b \pm \sqrt{D}}{2a}\).

Solving the quadratic equation further using the integer pairs helps verify that the solutions derived are real and fit the required conditions.