Question

complete the statements.
graph □ has one real root.
graph □ has a negative discriminant.
graph □ has an equation with coefficients a = 1, b = 4, c = -2
done

Explanation:

To solve this, we analyze each statement using properties of quadratic graphs (parabolas) and discriminant:

1. Graph with one real root

A quadratic has one real root when its graph touches the x - axis at exactly one point (vertex on the x - axis). From the graphs, Graph C (the left - most narrow parabola) touches the x - axis at one point. So Graph C has one real root.

2. Graph with negative discriminant

The discriminant of a quadratic \( ax^{2}+bx + c \) is \( D=b^{2}-4ac \). If \( D\lt0 \), the quadratic has no real roots (graph does not intersect the x - axis). Graph A (the right - most wider parabola) does not intersect the x - axis, so its discriminant is negative.

3. Graph with \( a = 1,b = 4,c=-2 \)

First, find the vertex of \( y=x^{2}+4x - 2 \). The x - coordinate of the vertex is \( x=-\frac{b}{2a}=-\frac{4}{2(1)}=-2 \). Substitute \( x = - 2 \) into the equation: \( y=(-2)^{2}+4(-2)-2=4 - 8 - 2=-6 \). The vertex is \( (-2,-6) \). Also, the discriminant \( D = 4^{2}-4(1)(-2)=16 + 8 = 24\gt0 \), so the graph intersects the x - axis at two points. Looking at the graphs, Graph B matches this (vertex position and two x - intercepts).

Final Answers

• Graph with one real root: \(\boldsymbol{C}\)
• Graph with negative discriminant: \(\boldsymbol{A}\)
• Graph with \( a = 1,b = 4,c=-2 \): \(\boldsymbol{B}\)

Answer:

To solve this, we analyze each statement using properties of quadratic graphs (parabolas) and discriminant:

1. Graph with one real root

A quadratic has one real root when its graph touches the x - axis at exactly one point (vertex on the x - axis). From the graphs, Graph C (the left - most narrow parabola) touches the x - axis at one point. So Graph C has one real root.

2. Graph with negative discriminant

The discriminant of a quadratic \( ax^{2}+bx + c \) is \( D=b^{2}-4ac \). If \( D\lt0 \), the quadratic has no real roots (graph does not intersect the x - axis). Graph A (the right - most wider parabola) does not intersect the x - axis, so its discriminant is negative.

3. Graph with \( a = 1,b = 4,c=-2 \)

First, find the vertex of \( y=x^{2}+4x - 2 \). The x - coordinate of the vertex is \( x=-\frac{b}{2a}=-\frac{4}{2(1)}=-2 \). Substitute \( x = - 2 \) into the equation: \( y=(-2)^{2}+4(-2)-2=4 - 8 - 2=-6 \). The vertex is \( (-2,-6) \). Also, the discriminant \( D = 4^{2}-4(1)(-2)=16 + 8 = 24\gt0 \), so the graph intersects the x - axis at two points. Looking at the graphs, Graph B matches this (vertex position and two x - intercepts).

Final Answers

• Graph with one real root: \(\boldsymbol{C}\)
• Graph with negative discriminant: \(\boldsymbol{A}\)
• Graph with \( a = 1,b = 4,c=-2 \): \(\boldsymbol{B}\)