The discriminant of an equation is 73. How many times will the graph of this equation touch the x–axis?
You are evidently talking about a quadratic equation of the form
y = Ax^2 + Bx + C = 0
There are two solutions (x values for which y=0) if the discriminant, B^2-4AC, is positive.
To determine the number of times the graph of an equation touches or intersects the x-axis, we need to consider the discriminant of the equation. The discriminant is a mathematical term used in quadratic equations and is denoted by the symbol Δ (delta).
For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant is given by the formula Δ = b^2 - 4ac. In this case, we are given that the discriminant is 73.
To solve for the discriminant, we can equate it to the given value of 73:
73 = b^2 - 4ac
From here, we can analyze the value of the discriminant to determine the number of times the graph of the equation touches the x-axis.
1. If the discriminant (Δ) is greater than zero (Δ > 0), then the equation has two distinct real roots. In this case, the graph of the equation intersects the x-axis at two different points, resulting in two touches or crossings.
2. If the discriminant (Δ) is equal to zero (Δ = 0), then the equation has exactly one real root. The graph of the equation touches the x-axis at a single point, resulting in one touch or crossing.
3. If the discriminant (Δ) is less than zero (Δ < 0), then the equation has no real roots. Consequently, the graph of the equation does not touch or intersect the x-axis at any real points.
Since we know that the discriminant is 73, which is greater than zero, we can conclude that the graph of this equation will touch the x-axis two times.