The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

ax^2 + bx + c = 1, where a, b, and c are constants and a > 0.

ax + bx^2 + c = 0, where a and b are coefficients.

ax^2 + bx = c, where a, b, and c are all positive.

x = (b ± √(b^2 - 4ac)) / (2c).

x = (b ± √(a^2 - 4bc)) / (2a).

The roots can be found using the formula: x = (-b ± √(b^2 - 4ac)) / (2a).

x = (-b ± √(b^2 - 4ca)) / a.

Only real and distinct.

The roots can be real and distinct, real and equal, or complex (non-real), depending on the value of the discriminant.

Real and equal or imaginary.

Only complex.

The discriminant is (b - 4ac) and determines the axis of symmetry.

The discriminant is b^2 - 4ac. It determines the nature of the roots: if >0, roots are real and distinct; if =0, roots are real and equal; if <0, roots are complex.

The discriminant is a^2 + 4bc.

The discriminant is 4ac - b. It determines if the roots exist.

If a quadratic can be factored into (px + q)(rx + s) = 0, the roots are x = -q/p and x = -s/r.

Factoring allows us to find the vertex of the parabola directly.

Factoring converts the equation into ax^2 + bx format for easier manipulation.

Factoring only determines the sign of the quadratic term.

Completing the square is used to rearrange the equation into bx^2 = c.

Completing the square involves rewriting ax^2 + bx + c in the form (x – h)^2 = k to solve for x.

It approximates roots by gaining a reciprocal form of the equation.

Use this technique to graph the function directly on a plane.

y = ax^2 + bx + c, where (h, k) represents intercepts.

Vertex form is y = (x – h)(x – k) + c.

The vertex form is y = a(x – h)^2 + k, where (h, k) is the vertex of the parabola.

y = a(x – h)^2 + b, where h is the maximum or minimum point.

The solutions (or roots) of the equation correspond to the x-intercepts of its graph.

Graphing finds the value of c directly.

Graphing helps identify symmetries instead of solutions.

It only identifies the vertex points.

The coefficient 'a' does not affect graph direction, only width.

The leading coefficient 'a' determines the direction of the parabola opening; if a > 0, it opens upwards, and if a < 0, it opens downwards.

Coefficient 'a' determines where the parabola intersects the y-axis.

The value of 'a' shows the number of real roots.

It is always x = b/2a, no matter the coefficients.

The axis is y = -b/(2a).

The axis of symmetry is x = -b/(2a).

It is determined by c alone.

Through multiplying both sides by b.

By using calculus to find turning points.

The quadratic formula is derived by completing the square on the general form ax^2 + bx + c = 0.

By directly equating the derivatives.

Yes, a quadratic equation can be solved by finding the x-intercepts of its graph.

No, graphing only estimates solutions.

Yes, but only if the discriminant is positive.

No, graphing is just for visualization purposes.

The sum of the roots, according to Vieta's formulas, is -b/a.

The sum is always b/a.

It depends on the discriminant.

The sum is determined by c and 'a'.

It varies depending on the value of 'b'.

The product is b/a.

The product of the roots, according to Vieta's formulas, is c/a.

The product is always zero for any quadratic.

Because 'a' only affects y-intercepts.

Because 'a' would make b a constant.

With a = zero, equation switches orientation.

If a were zero, the equation would not be quadratic, as ax^2 would be eliminated, reducing it to a linear equation.