Problem 31 - Contemporary Precalculus

Question

Find the number of real solutions of the equation by computing the discriminant. $$25 t^{2}+49=70 t$$

Step-by-Step Solution

Answer

Answer: The quadratic equation has 1 real solution.

Step 1: Identify the coefficients

In the given equation, 25t^2 - 70t + 49 = 0, we can identify the coefficients a, b, and c: a = 25 b = -70 c = 49

Step 2: Calculate the discriminant

Using the formula D = b^2 - 4ac, we will compute the discriminant: D = (-70)^2 - 4 * 25 * 49 D = 4900 - 4900 D = 0

Step 3: Determine the number of real solutions based on the discriminant

Since the discriminant D = 0, the equation has two real solutions which are equal. Therefore, this quadratic equation has 1 real solution.

Key Concepts

Quadratic Equation Solutions

Understanding the solutions to quadratic equations is a staple of algebra. Quadratic equations are of the form \( ax^2 + bx + c = 0 \) where 'a', 'b', and 'c' represent known numbers and 'a' does not equal zero. These equations always graph as a parabola (a U-shape curve) and their solutions are the 'x' values where this curve intersects the x-axis.

There can be one, two, or no real solutions to a quadratic equation. When there's exactly one solution, the parabola only touches the x-axis at a single point. This is known as a 'repeated root' or having a 'double root'. In the case of two solutions, the parabola crosses the x-axis at two distinct points. Finally, if the parabola does not intersect the x-axis at all, we say the equation has no real solutions; however, it may have complex solutions in this scenario.

Calculating Discriminants

The discriminant is the part of the quadratic formula \( \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) under the square root sign, given by \( b^2 - 4ac \) and denoted usually by 'D' or \( \Delta \). This value provides critical information about the nature of the solutions without solving the equation entirely.

If 'D' is positive, there are two distinct real solutions; if 'D' is zero, there is one real solution which is repeated; and if 'D' is negative, there are no real solutions, only complex ones. Calculating 'D' is a straightforward process: after identifying the coefficients 'a', 'b', and 'c', you square 'b' and subtract four times the product of 'a' and 'c'.

Quick Tip:

Remember that the coefficient 'a' will always multiply the squared term, 'b' the linear term, and 'c' is the constant.

Real Solutions of Quadratic Equations

The discriminant not only guides us in understanding the types but also the quantity of real solutions a quadratic equation holds. If the discriminant (D) is greater than zero, we have two distinct real solutions. When it equals zero, we have one unique real solution. This means our parabola touches the x-axis precisely at the vertex. Consequently, the quadratic has a perfect square, indicating that the same solution will work for both values of 'x' that satisfy the equation.

The absence of real solutions occurs when the discriminant is less than zero because you're left with the square root of a negative number, which falls into the realm of complex numbers. In our earlier problem, the discriminant was zero, signaling that there is just one real solution, which is also the apex where the parabola touches the x-axis - a situation sometimes described as the parabola 'kissing' the x-axis.