You're staring at a quadratic expression on a test, a homework assignment, or maybe a standardized exam. It reads: 6x² + 7x - 10. The question asks: *which statement about this expression is true?
And your mind goes blank.
Not because you don't know quadratics. The direction the parabola opens? The discriminant? So is it about the factors? But because "which statement is true" could mean anything. Still, the vertex? The roots? The y-intercept?
Here's the thing — most people freeze on this exact type of question because they try to evaluate every possible statement at once. The trick isn't knowing more math. It's knowing which math actually matters for the specific expression in front of you And that's really what it comes down to. Turns out it matters..
Let's break down 6x² + 7x - 10 completely. By the end, you'll be able to spot the true statement in any multiple-choice lineup — and explain why the others are wrong Which is the point..
What Is 6x² + 7x - 10
It's a quadratic expression. Standard form: ax² + bx + c.
In this case:
- a = 6 (the leading coefficient)
- b = 7 (the linear coefficient)
- c = -10 (the constant term)
That's it. No hidden tricks. But those three numbers determine everything about the expression — its shape, its roots, its vertex, its factorability, its graph Easy to understand, harder to ignore..
It's a trinomial with a leading coefficient ≠ 1
This matters more than you think. When a = 1, factoring is straightforward: find two numbers that multiply to c and add to b. That's why when a ≠ 1, you're playing a different game. You need two numbers that multiply to a × c (here, 6 × -10 = -60) and add to b (7).
That extra step trips up a lot of students. They try to factor it like a = 1 and get stuck.
The discriminant tells you the nature of the roots instantly
Discriminant = b² - 4ac.
Plug in: 7² - 4(6)(-10) = 49 + 240 = 289.
289 is a perfect square (17²). That means:
- Two real roots
- Two rational roots
- The quadratic factors nicely over the integers
If the discriminant weren't a perfect square, you'd have irrational roots. But if it were negative, no real roots at all. This one number — 289 — answers three major "true statement" categories before you even pick up a pencil Worth knowing..
Why It Matters / Why People Care
You're not factoring this for fun. You're factoring it because:
- Finding zeros lets you graph it, solve equations, analyze motion problems, optimize areas
- Factoring reveals the x-intercepts instantly
- Vertex form gives you the maximum or minimum value
- The discriminant tells you how many solutions exist without solving
In a multiple-choice question, the "true statement" is usually one of these:
- The vertex coordinates
- The solutions/roots/zeros
- On the flip side, the factored form
- The y-intercept
- In real terms, the axis of symmetry
- The direction of opening
Knowing all of them means you can verify any option in seconds Easy to understand, harder to ignore..
How It Works — Breaking Down Every Key Property
Factoring 6x² + 7x - 10
We need two numbers that multiply to a × c = -60 and add to b = 7.
List factor pairs of -60:
- 1 and -60 → sum -59
- 2 and -30 → sum -28
- 3 and -20 → sum -17
- 4 and -15 → sum -11
- 5 and -12 → sum -7
- 6 and -10 → sum -4
- 10 and -6 → sum 4
- 12 and -5 → sum 7 ← There it is
Rewrite the middle term using 12 and -5: 6x² + 12x - 5x - 10
Group: (6x² + 12x) + (-5x - 10)
Factor each group: 6x(x + 2) - 5(x + 2)
Factor out the common binomial: (6x - 5)(x + 2)
Factored form: (6x - 5)(x + 2)
Check: 6x·x = 6x², 6x·2 = 12x, -5·x = -5x, -5·2 = -10. 12x - 5x = 7x. ✓
Finding the Roots (Zeros, Solutions, x-Intercepts)
Set the factored form to zero: (6x - 5)(x + 2) = 0
Zero product property:
- 6x - 5 = 0 → 6x = 5 → x = 5/6
- x + 2 = 0 → x = -2
Roots: x = 5/6 and x = -2
These are rational, distinct, and real — exactly what the discriminant promised Nothing fancy..
Vertex and Axis of Symmetry
The vertex x-coordinate: **x = -b / (2a) = -7 / (2·6) = -
The vertex x‑coordinate is
[ x = -\frac{b}{2a}= -\frac{7}{2\cdot 6}= -\frac{7}{12}. ]
To find the y‑coordinate, substitute this value back into the original quadratic:
[ \begin{aligned} y &= 6\left(-\frac{7}{12}\right)^{2}+7\left(-\frac{7}{12}\right)-10 \ &= 6\cdot\frac{49}{144}-\frac{49}{12}-10 \ &= \frac{294}{144}-\frac{49}{12}-10 \ &= \frac{49}{24}-\frac{98}{24}-\frac{240}{24} \ &= -\frac{289}{24}. \end{aligned} ]
Thus the vertex is
[ \boxed{\left(-\frac{7}{12},;-\frac{289}{24}\right)}. ]
The axis of symmetry is the vertical line (x = -\frac{7}{12}). Because the leading coefficient (a = 6) is positive, the parabola opens upward, so the vertex represents the minimum point of the function.
The y‑intercept occurs at (x = 0):
[ y = 6(0)^{2}+7(0)-10 = -10, ]
giving the point ((0,-10)) Turns out it matters..
Quick‑check checklist for a “true statement” about (6x^{2}+7x-10)
| Property | Value / Form | How to verify in seconds |
|---|---|---|
| Factored form | ((6x-5)(x+2)) | Multiply out → (6x^{2}+7x-10) |
| Roots / zeros | (x = \frac{5}{6},; x = -2) | Set each factor to zero |
| Discriminant | (289 = 17^{2}) | (b^{2}-4ac) |
| Nature of roots | Two real, rational | Discriminant > 0 and perfect square |
| Vertex | (\left(-\frac{7}{12},-\frac{289}{24}\right)) | (x=-b/(2a)), plug in |
| Axis of symmetry | (x = -\frac{7}{12}) | Same as vertex x‑coordinate |
| y‑intercept | ((0,-10)) | Evaluate at (x=0) |
| Direction of opening | Upward (minimum) | (a=6>0) |
| Minimum value | (-\frac{289}{24}) ≈ (-12.04) | y‑coordinate of vertex |
Any multiple‑choice option that matches one of these entries can be confirmed instantly without re‑deriving the entire quadratic.
Conclusion
The quadratic (6x^{2}+7x-10) exemplifies how a single computation—the discriminant—unlocks a cascade of insights: it predicts the number and type of roots, guarantees a clean integer factorization, and leads directly to the vertex, axis of symmetry, and intercepts. By mastering these interconnected tools, students can move from tedious trial‑and‑error to rapid, reliable verification of any statement about a quadratic function. This fluency not only saves time on exams but also builds a deeper intuition for how algebraic expressions model real
Conclusion
The quadratic (6x^{2}+7x-10) exemplifies how a single computation—the discriminant—unlocks a cascade of insights: it predicts the number and type of roots, guarantees a clean integer factorization, and leads directly to the vertex, axis of symmetry, and intercepts. That said, this fluency not only saves time on exams but also builds a deeper intuition for how algebraic expressions model real‑world phenomena, from projectile motion to profit maximization. In real terms, by mastering these interconnected tools, students can move from tedious trial‑and‑error to rapid, reliable verification of any statement about a quadratic function. When faced with a quadratic, start with the discriminant—it’s often the key that opens every door.
Extension: Connecting to the Quadratic Formula
While factoring is efficient here, the quadratic formula (x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}) serves as the universal backup. Plugging in (a=6), (b=7), (c=-10) yields:
[ x = \frac{-7 \pm \sqrt{289}}{12} = \frac{-7 \pm 17}{12} ]
This produces the same roots: (x = \frac{10}{12} = \frac{5}{6}) and (x = \frac{-24}{12} = -2). In real terms, notice how the discriminant (\sqrt{289}=17) appears directly in the numerator; its status as a perfect square is the algebraic guarantee that the roots are rational and the trinomial factors over the integers. This reinforces a critical habit: always compute the discriminant first—it tells you whether factoring is even worth attempting.
Common Pitfalls to Avoid
- Sign errors in factoring: The factors ((6x-5)(x+2)) multiply to (+7x) because (12x + (-5x) = 7x). A common mistake is writing ((6x+5)(x-2)), which yields (-7x).
- Vertex arithmetic: The vertex (x)-coordinate is (-\frac{b}{2a} = -\frac{7}{12}). Forgetting the negative sign in the formula shifts the axis of symmetry to the wrong side of the (y)-axis.
- Minimum vs. Maximum: Because (a=6>0), the vertex is a minimum. Students often confuse the direction of opening when the linear coefficient (b) is positive, but only the sign of (a) determines concavity.
Final Conclusion
The quadratic (6x^{2}+7x-10) demonstrates that algebraic fluency is not about memorizing isolated formulas, but about recognizing the threads that connect them. By internalizing this checklist—discriminant, roots, factored form, vertex, intercepts—you transform a static equation into a dynamic mathematical object you can analyze, graph, and apply in seconds. On the flip side, the discriminant bridges the gap between the abstract (the nature of roots) and the concrete (factorability, vertex coordinates, graph shape). Whether you are optimizing a revenue function or calculating a projectile’s trajectory, this structured approach ensures you never miss a critical detail That's the whole idea..
