Identifying Standard Form

Is $3x^2 - 7x + 2 = 0$ a quadratic equation? If so, identify $a$, $b$, and $c$.

Check the degree: The highest power of $x$ is 2 (from $3x^2$) = Degree = 2

Verify $a \neq 0$: The coefficient of $x^2$ is 3, which is not zero = $a = 3 \neq 0$ ✓

Identify coefficients: Compare with $ax^2 + bx + c = 0$ = $a = 3$, $b = -7$, $c = 2$

Answer: Yes, it is a quadratic equation with $a = 3$, $b = -7$, and $c = 2$.

Rewriting in Standard Form

Rewrite $5x = x^2 + 6$ in standard form and identify the coefficients.

Move all terms to one side: Subtract $5x$ from both sides: $0 = x^2 - 5x + 6$ = $x^2 - 5x + 6 = 0$

Arrange in standard form: Already in form $ax^2 + bx + c = 0$ = $x^2 - 5x + 6 = 0$

Identify coefficients: Compare: $1 \cdot x^2 + (-5) \cdot x + 6 = 0$ = $a = 1$, $b = -5$, $c = 6$

Answer: $x^2 - 5x + 6 = 0$ with $a = 1$, $b = -5$, $c = 6$

Finding the Vertex

Find the vertex of the parabola $f(x) = x^2 - 4x + 3$.

Identify coefficients: $a = 1$, $b = -4$, $c = 3$ = Coefficients identified

Find x-coordinate of vertex: $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = \frac{4}{2} = 2$ = $x = 2$

Find y-coordinate: $f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$ = $y = -1$

State the vertex: The vertex is at $(x, y)$ = Vertex: $(2, -1)$

Answer: The vertex is at $(2, -1)$. Since $a = 1 > 0$, this is the minimum point of the parabola.

Analyzing Parabola Direction

For $f(x) = -2x^2 + 8x - 5$, determine whether the parabola opens upward or downward, and find its vertex.

Identify coefficient $a$: $a = -2$ = $a = -2 < 0$

Determine direction: Since $a < 0$, the parabola opens downward = Opens downward ∩

Find vertex x-coordinate: $x = -\frac{b}{2a} = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2$ = $x = 2$

Find vertex y-coordinate: $f(2) = -2(4) + 8(2) - 5 = -8 + 16 - 5 = 3$ = $y = 3$

Answer: The parabola opens downward with vertex at $(2, 3)$, which is the maximum point.

Mistake: Forgetting that $a$ cannot equal zero

Why: If $a = 0$, the equation becomes $bx + c = 0$, which is linear, not quadratic.

Correct: Always check that the coefficient of $x^2$ is non-zero before calling it quadratic.

Mistake: Sign errors when identifying $b$

Why: In $x^2 - 5x + 3 = 0$, students often say $b = 5$ instead of $b = -5$.

Correct: The coefficient includes its sign! Write: $x^2 + (-5)x + 3 = 0$, so $b = -5$.

Mistake: Confusing the vertex formula

Why: Students sometimes use $x = \frac{b}{2a}$ instead of $x = -\frac{b}{2a}$.

Correct: Remember: $x = -\frac{b}{2a}$ (negative sign in front!)

Mistake: Thinking all parabolas open upward

Why: The direction depends on the sign of $a$, not on any other coefficient.

Correct: If $a > 0$: opens upward (∪). If $a < 0$: opens downward (∩).

Projectile Motion

When you throw a ball, its height over time follows a quadratic equation.

A ball thrown upward has height $h = -5t^2 + 20t + 1.5$ meters after $t$ seconds. The vertex tells us the maximum height.

Business Profit

Companies use quadratic models to find the price that maximizes profit.

If profit is $P = -2x^2 + 200x - 3000$ where $x$ is the number of items sold, the vertex gives the optimal quantity.

Q: What's the difference between a quadratic equation and a quadratic function?

A: A quadratic equation is set equal to zero ($ax^2 + bx + c = 0$) and we solve for specific $x$ values. A quadratic function is written as $f(x) = ax^2 + bx + c$ and describes a relationship for all $x$ values.

Q: Why is the graph called a parabola?

A: The word comes from Greek 'parabole' meaning 'comparison' or 'application.' Mathematically, a parabola is defined as all points equidistant from a fixed point (focus) and a fixed line (directrix).

Q: Can a quadratic equation have no solutions?

A: A quadratic equation always has solutions, but they might be complex (imaginary) numbers. If the parabola doesn't cross the x-axis, the solutions are not real numbers.