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The discriminant is a key concept when dealing with quadratic equations. It's represented by the expression \( b^2 - 4ac \), which you find inside the square root of the quadratic formula. The discriminant helps determine the nature of the solutions of a quadratic equation.

• If the discriminant is positive, the quadratic equation has two distinct real number solutions.
• If the discriminant is zero, there is exactly one real number solution, which is also called a repeated root.
• If the discriminant is negative, the quadratic equation does not have real number solutions; instead, it has two complex solutions.

In the exercise, we calculated the discriminant for \( 7p^2 - 12p + 5 = 0 \) as \( 4 \), which is positive. This indicates that the quadratic equation has two distinct real number solutions. Knowing the discriminant's role can simplify the process by providing insight before fully solving the equation.

A quadratic equation is a type of polynomial equation of degree two, generally expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not zero. Such equations can graph as a parabola in a coordinate plane. The axis of symmetry and the vertex are important features of this parabola.

• "\( a \)" is the coefficient of the term \( x^2 \) and it determines the direction of the parabola. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
• "\( b \)" is the coefficient of the term \( x \) and affects the parabola's symmetry.
• "\( c \)" is the constant term and it places the vertex on the y-axis.

In our context, from the equation \( 7p^2 - 12p + 5 = 0 \), \( a = 7 \), \( b = -12 \), and \( c = 5 \). Understanding these components helps in using the quadratic formula to find the equation's solutions.

Real number solutions arise when the solutions of an equation are real numbers rather than complex numbers. When we use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), it is crucial to look at the discriminant to predict the nature of the solutions before diving into solving it.
Let's analyze the quadratic formula for real solutions:

• The formula will result in real solutions if the expression inside the square root, \( b^2 - 4ac \), is zero or positive.
• If \( \sqrt{b^2 - 4ac} \) equals \( 0 \), you'll get one real solution because the equation simplifies to just one value.
• If \( \sqrt{b^2 - 4ac} \) is a positive number, it contributes to "+/-," leading to two distinct real solutions.

In the exercise, since the discriminant was \( 4 \), a positive number, we confirmed that our equation, \( 7p^2 - 12p + 5 = 0 \), indeed had two distinct real number solutions: \( p = 1 \) and \( p = \frac{5}{7} \). These solutions can fully satisfy the original quadratic equation, making them real solutions. This concept is not only enlightening but also practical when you solve quadratic equations.