Hello,
I couldn't figure out how to solve the question below:
The equation of a curve is y = x^2 − 3x + 4.
Show that the whole of the curve lies above the x-axis.
Please help..
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How to show that a curve lies above the x-axis?
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Brent@GMATPrepNow
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Doesn't sound like a GMAT question to me.jspake wrote:Hello,
I couldn't figure out how to solve the question below:
The equation of a curve is y = x^2 − 3x + 4.
Show that the whole of the curve lies above the x-axis.
Please help..
What are the 5 answer choices?
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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kevincanspain
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Remember the quadratic formula ?
E = ax^2 + bx + c = 0 iff x= (-b +- sqrt(b^2 - 4ac))/2a
if b^2 - 4ac < 0, E=0 has no real roots and E and a have the same sign for all real numbers x
E = ax^2 + bx + c = 0 iff x= (-b +- sqrt(b^2 - 4ac))/2a
if b^2 - 4ac < 0, E=0 has no real roots and E and a have the same sign for all real numbers x
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GMAT/MBA Expert
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Brent@GMATPrepNow
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Since we're helping you with your homework (Algebra II perhaps?), here's another approach.jspake wrote:Hello,
I couldn't figure out how to solve the question below:
The equation of a curve is y = x^2 − 3x + 4.
Show that the whole of the curve lies above the x-axis.
Please help..
Use the Complete the Square technique.
y = x^2 − 3x + 4
y = x^2 − 3x + 2.25 - 2.25 + 4 (completed the square)
y = (x - 1.5)^2 - 2.25 + 4 (factored)
y = (x - 1.5)^2 + 1.75 (simplified)
So, this parabola has its vertex at coordinates (1.5, 1.75), which means the vertex is above the x-axis.
Since the parabola opens up (which I'll leave you to convince your teacher of
), this parabola must lie above the x-axis. Aside: For those studying for the GMAT (i.e., the vast majority of viewers), the Completing the Square technique is out of scope for the GMAT.
Cheers,
Brent
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savvysatyam
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Since a(the coefficient of quadratic equation) is greater than 0 and the determinant(b2)< 4ac, therfore it will be a parabola with face upwards and since determinant is not equal t zero it will be above x-axis and not touch it
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The Iceman
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Suppose we have a quadratic ax^2+bx+c=0
First, the sign of the coefficient of x^2 determines whether the parabola opens upwards or downwards. (a>0=> parabola opens up, and a<0=> parabola opens downwards)
Case 1: D=0 => Touches x axis at one point (significance: both roots real and equal)
Case 2: D>0 => Cuts x axis at two point (significance: the two roots are real and distinct)
Case 3: D<0 => Does not touch/cut/intercept x axis at any point (significance: both are non real roots)
First, the sign of the coefficient of x^2 determines whether the parabola opens upwards or downwards. (a>0=> parabola opens up, and a<0=> parabola opens downwards)
Case 1: D=0 => Touches x axis at one point (significance: both roots real and equal)
Case 2: D>0 => Cuts x axis at two point (significance: the two roots are real and distinct)
Case 3: D<0 => Does not touch/cut/intercept x axis at any point (significance: both are non real roots)
